P n
ilLD H
PLANE TRIGONOMETRY .
AND NUMERICAL COMPUTATION
/~\
^^^^^^
JOHN ALEXANDER JAMESON, Jr.
19031934
This book belonged to John Alexander Jameson, Jr., A.B., Wil
liams, 1925; B.S., Massachusetts Institute of Technology, 1928;
M.S., California, 1933. He was a member of Phi Beta Kappa, Tau
Beta Pi, the American Society of Civil Engineers, and the Sigma
Phi Fraternity. His untimely death cut short a promising career.
He was engaged, as Research Assistant in Mechanical Engineering,
upon the design and construction of the U. S. Tidal Model Labora
tory of the University of California.
His genial nature and unostentatious effectiveness were founded
on integrity, loyalty, and devotion. These qualities, recognized by
everyone, make his life a continuing beneficence. Memory of him
will not fail among those who knew him.
PLANE TRIGONOMETRY
AND NUMERICAL
COMPUTATION
BY
JOHN WESLEY YOUNG
»/
PROFESSOR OF MATHEMATICS
DARTMOUTH COLLEGE
AND
FRANK MILLETT MORGAN
AS8I8TANT PROFESSOR OF MATHEMATICS
DARTMOUTH COLLEGE
r°. .ss <+ Mi hi l p *c is l * * ■ ui
y If
Neta gork * *
THE MACMILLAN COMPANY
1919
All rights reserved
I C /> «
cat 1
l'£&f*' SAS35
Copyright, 1919,
Bv THE MACM1LLAN COMPANY.
Set up and electrotyped. Published October, 1919.
ENGINEERING Uftfttjty.
n.,1 p (AiJiXi.i ; ..!P,...' —
Norfoootr Pteaa / 1 * ^
.1. S_ Olshinor O.n 'Ro^T.rJnU JL Q^UU <^_
J. S. Cushing Co. — Berwick & Smith Co.
Norwood, Mass., U.S.A.
PREFACE
Ever since the publication of our Elementary Mathematical
Analysis (The Macmillan Co., 1917) we have been asked by
numerous teachers to publish separately, as a textbook in plane
trigonometry, the material on trigonometry and logarithms of
the text mentioned.
The present textbook is the direct outcome of these requests.
Of course, such separate publication of material taken out of
the body of another book necessitated some changes and an in
troductory chapter. As a matter of fact, however, we have
found it desirable to make a number of changes and additions
not required by the necessities of separate publication. As a
result fully half of the material has been entirely rewritten, with
the purpose of bringing the text abreast of the most recent
tendencies in the teaching of trigonometry.
There is an increasing demand for a brief text emphasizing the
numerical aspect of trigonometry and giving only so much of the
theory as is necessary for a thorough understanding of the
numerical applications. The material has therefore been ar
ranged in such a way that the first six chapters give the essen
tials of a course in numerical trigonometry and logarithmic
computation. The remainder of the theory usually given in
the longer courses is contained in the last two chapters.
More emphasis than hitherto has been placed on the use of
tables. For this purpose a table of squares and square roots
has been added. Recent experience has emphasized the appli
cations of trigonometry in navigation. We have accordingly
added some material in the text on navigation, have introduced
v
889757
vi PREFACE
the haversine, and have added a fourplace table of haversines
for the benefit of those teachers who feel that the use of the
haversine in the solution of triangles is desirable. This material
can, however, be readily omitted by any teacher who prefers
to do so.
J. W. Young,
F. M. Morgan.
Hanover, N.H.,
August, 1919.
CONTENTS
CHAPTER PAGES
I. Introductory Conceptions 110
II. The Right Triangle . . . . . . 1131
III. Simple Trigonometric Relations . . . . 3239
IV. Oblique Triangles . 4049
V. Logarithms 5060
VI. Logarithmic Computation 6174
VII. Trigonometric Relations 7587
VIII. Trigonometric Relations (continued) . . . 88103
Tables 106119
Index 121122
vu
PLANE TRIGONOMETRY AND
NUMERICAL COMPUTATION
CHAPTER I
INTRODUCTORY CONCEPTIONS
1. The Uses of Trigonometry. The word " trigonometry "
is derived from two Greek words meaning " the measurement
of triangles." A triangle has six socalled elements (or parts) •
viz., its three sides and its three angles.' AV'e v *rk s w from our
study of geometry that, in general, if three elements of a tri
angle (not all angles) are given, the triangle is completely
determined.* Hence, if three such determining elements of a
triangle are given, it should be possible to compute the remain
ing elements. The methods by which this can be done, i.e.
methods for " solving a triangle," constitute one of the prin
cipal objects of the study of trigonometry.
If two of the angles of a triangle are given, the third angle
can be found from the relation A + B f C = 180° (A, B, and
C representing the angles of the triangle) ; also, in a right tri
angle, if two of the sides are known, the third side can be
found from the relation a 2 + b 2 = c (a, b being the legs and c
the hypotenuse). But this is nearly the limit to which the
methods of elementary geometry will allow us to go in the
solution of a triangle.
Trigonometry f is the foundation of the art of surveying
* What exceptions are there to this statement ?
t Throughout this book we shall confine ourselves to the subject of "plane
trigonometry," which deals with rectilinear triangles in a plane. " Spherical
trigonometry" deals with similar problems regarding triangles on a sphere
whose sides are arcs of great circles.
B 1
PLANE TRIGONOMETRY
H,
and of much of the art of navigation. It is, moreover, of
primary importance in practically every branch of pure and
applied mathematics. Many of the more elementary applica
tions will be presented in later portions of this text.
2. The " Shadow Method." The ancient Greeks employed
the theory of similar triangles in the solution of a special type
of triangle problem which it is worth our while to examine
briefly, because it contains the germ of the theory of trigo
nometry.
It is desired to find the height CA of a vertical tower stand
ing on a level plain. It is observed
that at a certain time the tower casts a
shadow 42 ft. long. At the same time
a pole C'A', 10 ft. long, held vertically
with one end on the ground casts a
shadow 7 ft. long. From these data
the height of the tower is readily com
puted as follows : The right triangles
ABC and A'B'C are similar since Z B
= Z B'. (Why ?) Therefore we have
CA = C'A' 10
BC
A
A:
B' 7 C'
or
CA =
B'C
C'A'
B'C
The tower is then 60 ft. high.
3. A " Function " of an Angle.
£<7 = y x42 = 60.
From the point of view of
our future study the important thing to notice in the solution
CA C'A'
of the preceding article is the fact that the ratios , — —
v Bkj b g
are equal, i.e. that the ratio of the side opposite the angle B to
the side adjacent to the angle is determined by the size of the angle,
and does not depend at all on any of the other elements of the
triangle, provided only it is a right triangle.
I, § 3] INTRODUCTORY CONCEPTIONS 3
Definition. Whenever a quantity depends for its value on
a second quantity, the first is called a function of the second.
Thus in our example the ratio of the side opposite an angle
of a right triangle to the side adjacent is a quantity which
depends for its value only on the angle ; it is, therefore, called
a function of the angle. This ratio is merely one of several
functions of an angle which we shall define in the next
chapter. By means of these functions the fundamental prob
lem of trigonometry can be readily solved.
The particular function which we have discussed is called
the tangent of the angle. Explicitly defined for an acute angle
of a right triangle, we have
tangent of angle = ^ide op posite the ang le_.
side adjacent to the angle
If the angle B in the preceding example were measured it
would be found to contain 55°. In any right triangle then
containing an angle of 55° we should find this ratio to be equal
to  T , or 1.43. If the angle is changed, this ratio is changed,
but it is fixed for any given angle. If the angle is 45°, the
tangent is equal to 1, since in that case the triangle is
isosceles.
The word tangent is abbreviated " tan." Thus we have
already found tan 55° = 1.43 and tan 45° = 1.00. Similarly
to every other acute angle corresponds a definite number,
which is the tangent of that angle. The values of the tan
gents of angles have been tabulated. ^Ve shall have occasion
to use such tables extensively in the future. \
If a, 6, c are the sides of a right triangle ABC with right
angle at C and with the usual notation whereby the side a is
opposite the angle A and side b opposite the angle B, the defi
nition of the tangent gives
tanjB = .
a
PLANE TRIGONOMETRY
[I, §3
From this we get at once,
b = a tan B and a =
tan B
These are our first trigonometric formulas. By means of
them and a table of tangents we can compute either leg of a
right triangle, if the other leg and an acute angle are given.
EXERCISES
1. What is meant by "the elements of a triangle " ? by " solving a
triangle ' ' ?
2. A tree casts a shadow 20 ft. long, when a vertical yardstick with
one end on the ground casts a shadow of 2 ft. How high is the tree ?
3. A chimney is known to be 90 ft. high. How long is its shadow
when a 9foot pole held vertically with one end on the ground casts a
shadow 5 ft. long ?
4. Give examples from your own experience of quantities which are
functions of other quantities.
5. Define the tangent of an acute angle of a right triangle. Why does
its value depend only on the size of the angle ?
6. In the adjacent figure think of the line BA as rotating about the
point B in the direction of the arrow, starting from
the position BC (when the angle B is 0) and assum
ing successively the positions BA h BA%, BA 3 ,
Show that the tangent of the angle B is very
small when B is very small, that tan B increases as
the angle increases, that tan B is less than 1 as
long as B is less than 45°, that tan 45° = 1, that
tan B is greater than 1 if the angle is greater than
45°, and that tan B increases without limit as B ap
proaches 90°.
7. The following table gives the values of the tan
gent for certain values of the angle i
angle
10°
20°
30°
40°
50°
60°
70°
tangent
0.176
0.364
0.577
0.839
1.19
1.73
2.75
I, §4]
INTRODUCTORY CONCEPTIONS
//
(9)
60°
20°
By means of this table find the other leg of a right triangle ABC from
the elements given :
^ (a) B = 50°, a = 10 (d) B = 20°, b = 13
(6) B = 70°, a = 16 (e) A = 30°, 6=5
(c) B = 40°, & = 24 (/) A = 10°, & = 62
8. From the data and the results of the preceding exercise find the
other acute angle and the hypotenuse of each of the right triangles.
^4. Coordinates in a Plane. The student should already be
familiar from his study of algebra with the method of locating
points in a plane by means of coordinates. Since we shall
often have occasion to use such a method in the future, we will
recall it briefly at this point.
The method consists in referring the points in question to
two straight lines X'X and Y l Y, at right angles to each other,
which are called the axes of
Coordinates. X'X is USUally Second Quadrant
drawn horizontally and is
called the xaxis ; Y' Y, which
is then vertical, is called the
yaxis.
The position of any point
P is completely determined
if its distance (measured in
terms of some convenient
unit) and its direction from each of the axes is known. Thus
the position of P x (Fig. 2) is known, if we know that it is 4
units to the right of the ?/axis and 2 units above the xaxis. If
we agree to consider distance measured to the right or upwards
as positive, and therefore distance measured to the left or down
ward as negative ; and if, furthermore, we represent distances
and directions measured parallel to the xaxis by x, and distances
and directions measured parallel to the yaxis by y, then the
position of P x may be completely given by the specifications
» = r4, 2/=2; or more briefly still by the symbol (4, 2).
M,
X'
M s O
Third Quadrant
Mt
Mt
rl Fourth Quadrant
Fig. 2
6 PLANE TRIGONOMETRY [I, § 4
Similarly, the point P 2 in Fig. 2 is completely determined
by the symbol (3, 5). Observe that in such a symbol the x of
the point is written first, the y second. The two numbers x
and y, determining the position of a point, are called the
coordinates of the point, the x being called the xcoordinate
or abscissa, the y being called the ycoordinate or ordinate
of the point. What are the coordinates of P 3 and P A in
Fig. 2?
The two axes of coordinates divide the plane into four regions
called quadrants, numbered as in Fig. 2. The quadrant in
which a point lies is completely determined by the signs of its
coordinates. Thus points in the first quadrant are character
ized by coordinates (+, +), those in the second by ( — , +),
those in the third by ( — , — ), and those in the fourth by (f, — ).
Squareruled paper (socalled coordinate or cross section
paper) is used to advantage in " plotting " (i.e. locating) points
by means of their coordinates.
5. Magnitude and Directed Quantities. In the last article
we introduced the use of positive and negative numbers, i.e.
the socalled signed numbers, while in the preceding articles,
where we were concerned with the sides and angles of triangles,
we dealt only with unsigned numbers. The latter represent
magnitude or size only (as a length of 20 ft.), while the former
represent both a magnitude and one of two opposite direc
tions or senses (as a distance of 20 ft. to the left of a given
line). We are thus led to consider two kinds of quantities :
(1) magnitudes, and (2) directed quantities. Examples of the
former are : the length of the side of a triangle, the weight of
a barrel of flour, the duration of a period of time, etc. Ex
amples of the latter are : the coordinates of a point, the tem
perature (a certain number of degrees above or below zero),
the time at which a certain event occurred (a certain number
of hours before or after a given instant), etc.
I, § 6] INTRODUCTORY CONCEPTIONS 7
Geometrically, the distinction between directed quantities
and mere magnitudes corresponds to the fact that, on the one
hand, we may think of the line segment AB as drawn from A to
B or from B to A ; and, on the other hand, we
may choose to consider only the length of ' ' *~~* ' '
such a segment, irrespective of its direction.
Figure 3 exhibits the geometric representation
of 5, + 5, and — 5. A segment whose direc
tion is definitely taken account of is called ^'directed segment.
The magnitude of a directed quantity is called its absolute
value. Thus the absolute value of — 5 (and also of + 5) is 5.
Observe that the segments OM u M X P X (Fig. 2) representing
the coordinates of P x are directed segments.
6. Directed and General Angles. In elementary geometry
an angle is usually defined as the figure formed by two half
lines issuing from a point. However, it is often more serviceable
to think of an angle as being generated
by the rotation in a plane of a halfline
OP about the point as a pivot, start
ing from the initial position OA and
ending at the terminal position OB (Fig.
4). We then say that the line OP has
generated the angle AOB. Similarly, if OP rotates from the
initial position OB, to the terminal position OA, then the angle
BOA is said to be generated. Considerations similar to those
regarding directed line segments (§ 5) lead us to regard one of
the above directions of rotation as positive and x the other as
negative. It is of course quite immaterial which one of the
two rotations we regard as positive, but
we shall assume, from now on, that
counterclockwise rotation is positive and
clockwise rotation is negative.
Still another extension of the notion Fig. 5
8
PLANE TRIGONOMETRY
[I, §6
of angle is desirable. In elementary geometry no angle greater
than 360° is considered and seldom one greater than 180°. But
from the definition of an angle just given, we see that the
revolving line OP may make any number of complete revolu
tions before coming to rest, and thus the angle generated may
be of any magnitude. Angles generated in this way abound
in practice and are known as angles of rotation *
When the rotation generating an angle is to be indicated, it is
customary to mark the angle by means of an arrow starting at
the initial line and ending at the terminal line. Unless some
such device is used, confusion is liable to result. In Fig. G
30°
390'
750
1110
Fig. (5
angles of 30°, 390°, 750°, 1110°, are drawn. If the angles were
not marked one might take them all to be angles of 30°.
7. Measurement of Angles. For the present, angles will be
measured as in geometry, the degree (°) being the unit of measure. A
complete revolution is 360°. The other units in this system are the
minute ('), of which 60 make a degree, and the second ("), of which 60
make a minute. This system of units is of great antiquity, having been
used by the Babylonians. The considerations of the previous article then
make it clear that any real number, positive or negative, may represent an
angle, the absolute value of the number representing the magnitude of
the angle, the sign representing the direction of rotation.
v
Fig. 7
Consider the angle XOP = 0, whose vertex O coincides with the origin
of a system of rectangular coordinates, and whose initial line OX coin
*For example, the minute hand of a clock describes an angle of —180°
n 30 minutes, an angle of — 540° in 90 minutes, and an angle of — 720° in 120
ninutes.
I, § 8] INTRODUCTORY CONCEPTIONS 9
cides with the positive half of the a;axis (Fig. 7) . The angle is then
said to be in the first, second, third, or fourth quadrant, according as its
terminal .line OP is in the first, second, third, or fourth quadrant.
8. Addition and Subtraction of Directed Angles. The
meaning to be attached to the sum of two directed angles is analogous to
that for the sum of two directed
line segments. Let a and b be /* /&
two halflines issuing from the / £,
Y
same point O and let (ab) repre
sent an angle obtained by rotat j£F SJ — ' 5~ q
ing a half line from the position jr IG# y
a to the position b. Then if we
have two angles (a&) and (6c) with the same vertex O, the sum (a6) + (6c)
of the angles is the angle represented by the rotation of a half line from
the position a to the position b and then rotating from the position b to the
position c. But these two rotations are together equivalent to a single rota
tion from a to c, no matter what the relative positions of a, 6, c may have
been. Hence, we have for any three half lines a, b, c issuing from a point 0,
(1) (ab) + (bc)=(ac), (ob) + (bc)=0, (ab) = (cb)(ca).
It must be noted, however, that the equality sign here means " equal,
except possibly for multiples of 360V The proof of the last relation is >
left as an exercise. ^^
EXERCISES \^\)
1. On squareruled paper draw two axes of reference and then plot the
following points: (2, 3), ( 4, 2), ( 7,  1), (0,  3), (2,  5), (5, 0).
2. What are the coordinates of the origin ?
3. Where are all the points for which x — 2? x =— 3 ? y — — 1 ?
y = ±? x = 0?
4. Show that any point P on the 2/axis has coordinates of the form
(0, y) . What is the form of the coordinates of any point on the xaxis ?
5. A right triangle has the vertex of one acute angle at the origin and
one leg along the seaxis. The vertex of the other acute angle is at
(7, 10). What is the tangent of the angle at O ? *? \
6. What angle does the minute hand of a clock describe in 2 hours
and 30 minutes ? in 4 hours and 20 minutes ? ' \ / a
7. Suppose that the dial of a clock is transparent so that it may be
read from both sides. Two persons stationed at opposite sides of the dial
observe the motion of the minute hand. In what respect will the angles
described by the minute hand as seen by the two persons differ?
10 PLANE TRIGONOMETRY [I, § 8
i X /
4 8. In what quadrants are the following angles : 87° ? 135° ? — 325° ?
540°? 1500°? 270°?
9. In what quadrant is 0/2 if is a positive angle less than 360° and in
the second quadrant ? third quadrant ? fourth quadrant ?
10. By means of a protractor construct 27° + 85° + (— 30°) + 20° +
(45°).
11. By means of a protractor construct — 130° + 56° — 24°.
I
J
CHAPTER II
THE RIGHT TRIANGLE
9. Introduction. At the beginning of the preceding chap
ter we described the fundamental problem of trigonometry to
be the " solution of the triangle," i.e. the problem of com
puting the unknown elements of a triangle when three of the
elements (not all angles) are given. This problem can be
solved by finding relations between the sides and angles of a
triangle by means of which it is possible to express the un
known elements in terms of the known elements. In order
to establish such relations, it has been found desirable to
define certain functions of an angle. One such function — the
tangent — was introduced in § 3 by way of preliminary illus
tration.
In the present chapter, we shall give a new definition of the
tangent of an angle and also define two other equally impor
tant functions — the sine and the cosine. It should be noted
that the definition given for the tangent in § 3 applies only to
an acute angle of a right triangle. For the purposes of a sys
tematic study of trigonometry we require a more general defini
tion, which will apply to any angle, positive or negative, and
of any magnitude. Such definitions are given in the next
article, in which the notion of a system of coordinates plays a
fundamental role, the notion of a triangle not being introduced
at all. After considering some of the consequences of our
definitions in §§ 1113, we consider the way in which these
definitions enable us to express relations between the sides
and angles of a right triangle. These results are then imme
diately applied to the solution of numerical problems by means
r of tables and to applications in surveying and navigation.
11
12
PLANE TRIGONOMETRY
[II, § 10
10. The Sine, Cosine, and Tangent of an Angle. We
may now define three of the functions referred to in § 3. To
this end let = XOP (Fig. 9) be any directed angle, and let
zyL
us establish a system of rectangular coordinates in the plane
of the angle such that the initial side OX of the angle is the
positive half of the scaxis, the vertex being at the origin and
the yaxis being in the usual position with respect to the
#axis. Let the units on the two axes be equal. Finally, let
P be any point other than on the terminal side of the angle
6, and let its coordinates be (x, y). The directed segment
OP = r is called the distance of P and is always chosen posi
tive. The coordinates x and y are positive or negative accord
ing to the conventions previously adopted. We then define
The sine of 8 =
The cosine of 6 =
ordinate of P _ y
distance of P ~ r
abscissa of P x
distance of P
™* , * /v ordinate of P y . . _
The tangent of 8 = r — . jp=~, provided x =£ 0.*
These functions are usually written in the abbreviated forms
sin 0, cos 0, tan 0, respectively ; but they are read as " sine 0"
" cosine 0," " tangent 0." It is very important to notice that
the values of these functions are independent of the position
of the point P on the terminal line. For let P' (x\ y') be any
other point on this line. Then from the similar right triangles
xyrf and x'y'r 1 it follows that the ratio of any two sides
of the triangle xyr is equal in magnitude and sign to the
* Prove that x and y cannot be zero simultaneously.
t Triangle xyz means the triangle whose sides are x, y, z.
II, § 11]
THE RIGHT TRIANGLE
13
ratio of the corresponding sides of the triangle x'y'r'. There
fore the values of the functions just defined depend merely
on the angle 9. They are onevalued functions of 6 and are
called trigonometric functions.
Since the values of these functions are defined as the ratios
of two directed segments, they are abstract numbers. They
may be either positive, negative, or zero. Remembering that r
is always positive, we may readily verify that the signs of the
three functions are given by the following table.
Quadrant
Sine
Cosine
Tangent
1
• +
+
2
3
+
4
+
11. Values of the Functions for 45°, 135°, 225°, 315°. In
each of these cases the triangle xyr is isosceles. Why?
Since the trigonometric functions are independent of the
position of the point P on the terminal line, we may choose
the legs of the right triangle xyr to be of length unity, which
M. &i
C^L
'^%\
Fig. 10
gives the distance OP as V2. Figure 10 shows the four angles
with all lengths and directions marked. Therefore,
1
sin 45°= ,
V2
cos 45° =
sin 135° = — ,
V2
cos 135° =
sin 225° = — ,
V2
cos 225° =
sin 315° = —,
cos 315° =
V2
1
i
i
V2
tan 45° = 1,
tan 135° = 1,
tan 225° = 1,
tan 315° =  1.
14
PLANE TRIGONOMETRY
[II, § 12
12. Values of the Functions for 30°, 150°, 210°, 330°. From
geometry we know that if one angle of a right triangle con
tains 30°, then the hypotenuse is double the shorter leg,
which is opposite the 30° angle. Hence if we choose the
shorter leg (ordinate) as 1, the hypotenuse (distance) is 2,
Ml 'I<s^L
vz
•vT
dLL±
t»
Fig. 11
and the other leg (abscissa) is V3. Figure 11 shows angles of
30°, 150°, 210°, 330° with all lengths and directions marked.
Hence we have
cos 30°=^, tan 30* = — ,
2 ' V3
sin W;,
sin 150° = ^,
sin 210° =
2'
sin 330° =  ,
2'
cos 150° =  ^?, tan 150° =  — ,
2 V3
cos 210° =
V3
2 '
cos 330 c
V3
2 :
tan 210° =
V3 J
tan 330° = 
V3
13. Values of the Functions for 60°, 120°, 240°, 300°. It is
left as an exercise to construct these angles and to prove that
sin 60° = ^5,
cos 60 c
sin 120° = ^,
cos 120° = ,
2'
sin 240°=^?,
. 2 '
cos 240° = ,
2'
sin 300° = ^,
2
cos 300° =1,
tan 60°=V3,
tanl20° = V3,
tan240°=V3,
tan 300° =  V3.
II, § 14]
THE RIGHT TRIANGLE
15
14. Sides and Angles of a Right Triacgle. Evidently any
right triangle ABC can be so placed in a system of coordi
nates that the vertex of either acute
angle coincides with the origin O
and that the ad'jacent leg lies along
the positive end OX of the ajaxis
(Fig. 12). The following relations
then follow at once from the defini
tions of the sine, cosine, and tangent
of § 10.
In any right triangle, the trigonometric functions of either acute
angle are given by the ratios :
the sine
the cosine =
side opposite the angle
hypotenuse
side adjacent to the angle
the tangent
hypotenuse
side opposite the angle
side adjacent to the angle '
These relations are fundamental in all that follows. They
should be firmly fixed in mind in such a way that they can be
readily applied to any right triangle in what
ever position it may happen to be (for example
as in Fig. 13). The student should be able to
reproduce any of the following relations with
out hesitation whenever called for. They
should not be memorized, but should be read
from an actual or imagined figure :
b
Fig. 13
sin^l
cos A
sin B
cos B=,
c
tan A =  , tan B =
Also the known relation :
C 2 = a 2 + b 2 .
16
PLANE TRIGONOMETRY
[II, § 14
If any two elements (other than the right angle) of a right
triangle are given, we can then find a relation connecting these
two elements with any unknown element, from which relation
the unknown element can be computed.
15. Applications. The angle which a line from the eye to
an object makes with a horizontal line in the same vertical
plane is called an angle of elevation or an angle of depression,
Horizontal
Fig. 14
according as the object is above or below the eye of. the ob
server (Fig. 14). Such angles occur in many examples.
Example 1. A man wishing to know the distance between two points
A and B on opposite sides of a pond locates a point C on the land (Fig.
15) such that AC = 200 rd., angle C = 30°, and angle B = 90°. Find the
distance AB.
AB
AG
AB = AC sin G
= 200 • sin 30°
100 rd.
Solution :
sin C. (Why ?)
= 200 • *
Fig. 15
Example 2. Two men stationed at points A and G 800 yd. apart and
in the same vertical plane with a balloon B, observe simultaneously the
angles of elevation of the balloon to be 30° and 45° respectively. Find the
height of the balloon.
Solution : Denote the height of the balloon DB by y, and let DC = x;
then AD = 800  x.
L
800x D x
Fig. 16
II, § 15J THE RIGHT TRIANGLE 17
Since tan 45° = 1, we have 1 =,
x
1 y
and since tan 30° =s 1/V3, we have —  == — g ^ _ x '
Therefore x = y and 800 — x — y V3.
800
Solving these equations for y, we have y — = 292.8 yd.
V3 + 1
EXERCISES
• 1. In what quadrants is the sine positive ? cosine negative ? tangent
positive ? cosine positive? tangent negative ? sine negative ?
2. In what quadrant does an angle lie if
(a) its sine is positive and its cosine is negative ?
(6) its tangent is negative and its cosine is positive?
(c) its sine is negative and its cosine is positive ?
(d) its cosine is positive and its tangent is positive ?
3. Which of the following is the greater and why : sin 49° or cos 49° ?
£in 35° or cos 35° ?
4. If 6 is situated between 0° and 360°, how many degrees are there in
6 if tan = 1? Answer the similar question for sin = % ; tan $ = — 1 .
5. Does sin 60° = 2 • sin 30° ? Does tan 60° = 2 • tan 30° ? What
can you say about the truth of the equality sin 2 = 2 sin 6 ?
M) The Washington Monument is 555 ft. high. At a certain place in
the plane of its base, the angle of elevation of the top is 60°. How far is
that place from the foot and from the top of the tower ?
— "^. A boy whose eyes are 5 ft. from the ground stands 200 ft. from a
flagstaff. From his eyes, the angle of elevation of the top is 30°. How
high is the flagstaff ?
8. A tree 38 ft. high casts a shadow 38 ft. long. What is the angle
of elevation of/the top of the tree as seen from the end of the shadow ?
How far is i*4rom the end of the shadow to the top of the tree ?
i'rom the top of a tower 100 ft. high, the angle of depression of
two stones, which are in a direction due east and in the plane of the base
are 45° and 30° respectively. How far apart are the stones ?
.4ns. 100( V3  1) = 73.2 ft.
18
PLANE TRIGONOMETRY
[II, § 15
10. Find the area of the isosceles triangle in which the equal sides 10
inches in length include an angle of 120°. Ans. 25 V3 = 43.3 sq. in.
^11. Is the formula sin 2 = 2 sin cos true when = 30° ? 60° ?
120°?
<l2! From a figure prove that sin 117° = cos 27°.
13. Determine whether each of the following formulas is true when
= 30°, 60°, IHS , 210 D :
1 + tan 2 = —  —
COS 2 '
1 +  1  — *,
tan 2 sin 2
sin 2 f cos 2 i
1.
,""i4. Let Pi(Xi, ?/i) and Pz(x2, yt) be any two points the distance be
tween which is r (the units on the axes being equal) . If is the angle
that the line PiP% makes with the xaxis, prove that
x 2  Xi , ?/2
*r=^» = 2 r.
}l6. Computation of the Value of One Trigonometric
Function from that of Another.
J>±£Si
Fig. 17
Example 1. Given that sin = f, find the
values of the other functions.
Since sin is positive, it follows that is
an angle in the first or in the second quad
rant. Moreover, since the value of the sine
is , then y = 3 • k and r = 5 • k, where k is
any positive constant different from zero. (Why?) It is, of course,
immaterial what positive value we assign to k, so we shall assign the
value 1. We know, however, that the abscissa, ordinate, and distance
are connected by the relation x 2 + y 2 = r 2 , and hence it follows that
x = ± 4. Figure 17 is then selfexplanatory. Hence we have, for the first
quadrant, sin = f , cos = f , and tan = £ ; for the second quadrant,
sin = , cos = — , tan = — f .
is negative, find the other trigonometric functions of
the angle 0.
Since sin is positive and tan is negative, must
be in the second quadrant. We can, therefore, con
struct the angle (Fig. 18), and we obtain sin = ^ T ,
cos = — Y§, tan = — T \.
Fig. 18
II, § 17]
THE RIGHT TRIANGLE
19
k
17. Computation for Any Angle. Tables. The values of
the trigonometric functions of any angle may be computed by
the graphic method. For
example, let us find the
trigonometric functions of
35°. We first construct
on squareruled paper,
by means of a protractor,
an angle of 35° and choose
a point P on the ter
minal line so that OP
shall equal 100 units.
Then from the figure we
find that 0^=82 units
and MP = 57 units.
Therefore
TOT
_L'.
;
::
l >0~
TPT
■':':':
[jjT
■■:■;■■
V*
:: : rj:::i
b :::
.ft ■•■
■to — : —
■i'>:.\.'..;.
A :
' •
:: :i\
m
V
*
10 XV SO 40 SO 60 70 (SO 90 100
Fig. 19
sin 35° = tVv = ° 57 > cos 35 ° = Tiro = ° 82 > tan 35 ° = U = 0.70.
The tangent may be found more readily if we start by tak
ing OA = 100 units and then measure AB. In this case,
AB = 70 units and hence tan3o° = ^^ = 0.70.
It is at once evident that the graphic method, although
simple, gives only an approximate result. However, the values
of these functions have been computed accurately by methods
beyond the scope of this book. The results have been put in
tabular form and are known as tables of natural trigonometric
functions. Such tables and how to use them will be discussed
in the next article.
Figure 20 makes it possible to read off the sine, cosine, or
tangent of any angle between 0° and 90° with a fair degree of
accuracy. The figure is selfexplanatory. In reading off
values of the tangent use the vertical line through 100 for angles
up to 55°, and the line through 10 for angles greater than 55°.
Its use is illustrated in some of the following exercises.
20
PLANE TRIGONOMETRY
[II, § 17
10
Fig
so to
20. — Graphical, T
60 60 70 60 90 100
able oe Trigonometric Functions
II, § 18] THE RIGHT TRIANGLE 21
EXERCISES
Find the other trigonometric functions of the angle 6 when
t£)tan0 = 3. 3. cos = 1$. 5. sin0 = f.
2. sin0 = . 4. tan0=f 6. cos0= — .
rl) sin = f and cos is negative.
8. tan = 2 and sin is negative.
9. sin = — \ and tan is positive.
10. cos = § and tan is negative.
11. Can 0.6 and 0.8 be the sine and cosine, respectively, of one and
the same angle ? Can 0.5 and 0.9 ? Ans. Yes ; no.
12. Is there an angle whose sine is 2 ? Explain.
13. Determine graphically the functions of 20°, 38°, 70°, 110°.
14. From Fig. 20, find values of the following :
sin 10°, cos 50°, tan 40°, sin 80°, tan 70°, cos 32°, tan 14°, sin 14°.
15. A tower stands on the shore of a river 200 ft. wide. The angle of
elevation of the top of the tower from the point on the other shore exactly
opposite to the tower is such that its sine is \. Find the height of the
tower.
16. From a ship's masthead 160 feet above the water the angle of de
pression of a boat is such that the tangent of this angle is / 2 . Find the
distance from the boat to the ship. Ans. 640 yards.
18. Use of Tables of Trigonometric Functions. Examina
tion of the tables of " Four Place Trigonometric Functions "
(p. 112) shows columns headed " Degrees," " Sine," " Tangent,"
" Cosine," and under each of the last three named a column
headed " Value " (none of the other columns eoncern us at pres
ent). Two problems regarding the use of these tables now
present themselves.
1. To find the value of a function when the angle is given.
(a) Find the value of sin 15° 20'. In the column headed
" Degrees " locate the line corresponding to 15° 20' (p. 113) ; on
the same line in the " value " column for the " Sine," we read
the result : sin 15° 20' = 0.2644. On the same line, by using
the proper column, we find tan 15° 20' = 0.2742, and cos 15° 20'
= 0.9644.
/ H
22 PLANE TRIGONOMETRY [II, § IS
(b) Find the value of tan 57° 50'. The entries in the
column marked " Degrees " at the top only go as far as 45°
(p. 116). But the columns marked " Degrees " at the bottom
contain entries beginning with 45° (p. 116) and running back
wards to 90° (p. 112). In using these entries we must use the
designations at the bottom of the columns. Thus on the line
corresponding to 57° 50' (p. 115) we find the desired value :
tan 57° 50' = 1.5900. Also sin 57° 50' = 0.8465, and cos 57°
50' = 0.5324.
(c) Find the value of sin 34° 13'. This value lies between
the values of sin 34° 10' and 34° 20'. We find for the latter
sin 34° 10' = 0.5616
sin 34° 20' = 0.5640
Difference for 10' = 0.0024
Assuming that the change in the value of the function
throughout this small interval is proportional to the change in
the value of the angle, we conclude that the change for 1' in the
angle would be 0.00024. For 3', the change in the value of the
function would then be 0.00072. Neglecting the 2 in the last
place (since we only use four places and the 2 is less than 5),
we find sin 34° 13' = 0.5616 + 0.0007 = 0.5623. This process is
called interpolation. With a little practice all the work in
volved can and should be done mentally ; i.e. after locating the
place in the table (and marking it with a finger), we observe
that the " tabular difference " is " 24 " ; we calculate mentally
that .3 of 24 is 7.2, and then add 7 to 5616 as we write down
the desired value 0.5623.
Similarly we find tan 34° 13' = 0.6800 (the correction to be
added is in this case 12.9 which is " rounded off " to 13) and
cos 34° 13' = 0.8269. (Observe that in this case the correction
must be subtracted. Why ?)
2. To find the angle when a value of a function is given.
ere we proceed in the opposite direction. Given sin A =
■J
N
II, § 18] THE RIGHT TRIANGLE 23
0.3289 ; find A. An examination of the sine column shows
that the given value lies between sin 19° 10' ( = 0.3283) and.
> sin 19° 20'(= 0.3311). We note the tabular difference to be 28.
The correction to be applied to 19° 10' is then fa of 10' = f f '
=  1 /' = 2.1'. Hence A = 19° 12.1'. (With a four place table
do not carry your interpolation farther than the nearest tenth
of a minute.) (See § 20.) \
EXERCISES
* 1. For practice in the use of tables, verify the following :
(a) sin 18° 20' = 0.3145 (d) sin 27° 14' =0.4576 (g) sin 62° 24M =0.8862
(6) cos 37° 30' =0.7934 (e) cos 34° 11' =0.8272 (h) cos 59° 46' .2 =0.5034
(c) tan 75° 50' =3.9617 (/) tan 68° 21' = 2.5173 (i) tan 14° 55'.6 =0.2665
Assume first that the angles are given and verify the values of the
functions. Then assume the values of the functions to be given and
verify the angles.
2. A certain railroad rises 6 inches for every 10 feet of track. What
angle does the track make with the horizontal ?
NJ
3. On opposite shores of a lake are two flagstaffs A and B. Per
pendicular to the line AB and along one shore, a line BC = 1200 ft. is
measured. The angle ACB is observed to be 40° 20'. Find the distance
between the two flagstaffs.
4. The angle of ascent of a road is 8°. If a man walks a mile up the
road, how many feet has he risen ?
\
5. How far from the foot of a tower 150 feet high must an observer,
6 ft. high, stand so that the angle of elevation of its top may be 23°. 5 ?
6. From the top of a tower the angle of depression of a stone in the
lane of the. base is 40° 20'. What is the angle of depression of the stone
from a point halfway down the tower?
7. The altitude of an isosceles triangle is 24 feet and each of the equal
angles contains 40° 20'. Find the lengths of the sides and area of the
triangle.
8. A flagstaff 21 feet high stands on the top of a cliff. From a point
on the level with the base of the cliff, the angles of elevation of the top
and bottom of the flagstaff are observed. Denoting these angles by «
and /3 respectively, find the height of the cliff in case sin a = / 7 and
Ans. 75 feet.
\
24 PLANE TRIGONOMETRY [II, § 18
9. A man wishes to find the height of a tower CB which stands on a
horizontal plane. From a point A on this plane he finds the angle of ele
vation of the top to be such that sin CAB = f . From a point A' which
is on the line AC and 100 feet nearer the tower, he finds the angle of
elevation of the top to be such that tan CA'B'= §. Find the height of
fche tower.
10. Find the radius of the inscribed and circumscribed circle of a regu
ar pentagon whose side is 14 feet.
11. If a chord of a circle is two thirds of the radius, how large an
angle at the center does the chord subtend ?
19. Computation with Approximate Data. Significant
Figures. The numerical applications of trigonometry (in sur
veying, navigation, engineering, etc.) are concerned with com
puting the values of certain unknown quantities (distances,
angles, etc.) from known data which are secured by measure
ment. Now, any direct measurement is necessarily an approxi
mation. A measurement may be made with greater or less
accuracy according to the needs of the problem in hand — but
it can never be absolutely exact. Thus, the information on a
signpost that a certain village is 6 miles distant merely
means that the distance is 6 miles to the nearest mile — i.e. that
the distance is between 5± and 6^ miles. Measurements in a
physical or engineering laboratory need sometimes to be made to
the nearest one ' thousandth of an inch. For example the bore
of an engine cylinder may be measured to be 3.496 in., which
means that the bore is between 3.4955 in. and 3.4965 in.
A simple convention makes it possible to recognize at a
glance the degree of accuracy implied by a number represent
ing an approximate measure (either direct or computed). This
convention consists simply in the agreement to write no more
figures than the accuracy warrants. Thus in arithmetic 6 and
6.0 and 6.00 all mean the same thing. This is not so, when
these numbers are used to express the result of measurement
or the result of computation from approximate data. Thus 6
means that the result is accurate to the nearest unit, 6.0 that
II, § 20] THE RIGHT TRIANGLE 25
it is accurate to the nearest tenth of a unit, 6.00 to the nearest
hundredth of a unit.
These considerations have an important bearing on practical
computation. If the side of a square is measured and found
to he 3.6 in. and the length of the diagonal is computed by
the formula : diagonal =/ side x V2^4t would be wrong to write
= 3.6 x V2 = 3.6 x 1.4142 = 5.09112 in. The correct result
is 5.1 in. For the computed value of the diagonal cannot be
more accurate than the measured value of the side. The result
5.09112 must therefore be " rounded off " to two significant
figures, which gives 5.1. As a matter of fact for the purpose
of this problem V2 = 1.4142 should be rounded before multi
plication to V2 = 1.4 ; thereby reducing the amount of labor
necessary.
A number is " rounded off," by dropping one or more digits
at the right and, if the last digit dropped is 5 + , 6, 7, 8, or 9
increasing the preceding digit by 1.* Thus the successive
approximations to w obtained by rounding of 3.14159 ••• are
3.1416, 3.142, 3.14, 3.1, 3.
20. The Number of Significant Figures of a number (in the
decimal notation) may now be defined as the total number of
digits in the number, except that if the number has no digits
to the right of the decimal point, any zeros occurring between
the decimal point and the first digit different from zero are
not counted as significant. Thus, 34.06 and 3,406,000 are both
numbers of four significant figures : while 3,406,000.0 is a
number of eight significant figures.!
* In rounding off a 5 computers round off to an even digit. Thus 1.415
would be rounded to 1.42, whereas 1.445 would be rounded to 1.44. If this
rule is used consistently the errors made will tend to compensate each other.
t Confusion will arise in only one case. For example, if 3999.7 were
rounded by dropping the 7 we should write it as 4000 which according to the
above definition would have only 1 significant figure, whereas we know from
the way it was obtained that all four figures are significant. In such a case
we may underscore the zeros to indicate they are significant or use some
other device.
26 PLANE TRIGONOMETRY [II, § 20
In any computation involving multiplication or division the
number of significant figures is generally used as a measure of
the accuracy of the data. A computed result should not in
general contain more significant figures than the least accurate
of the data. But computers generally retain one additional
figure during the computation and then properly round off the
final result. Even then the last digit may be inaccurate — but
that is unavoidable.
The following general rules will be of use in determining
the degree of accuracy to be expected and in avoiding useless
labor :
1. Distances expressed to two significant figures call for
angles expressed to the nearest 30' and vice versa.
2. Distances expressed to three significant figures call for
angles expressed to the nearest 5', and vice versa.
3. Distances expressed to four significant figures call for
angles expressed to the nearest minute, and vice versa.
4. Distances expressed to five significant figures call for
angles expressed to the nearest tenth of a minute, and vice,
versa.
In working numerical problems the student should use every
safeguard to avoid errors. Neatness and systematic arrange
ment of the work are important in this connection. All work
should be checked in one or more of the following ways.
1. Gross errors may be detected by habitually asking oneself :
Is this result reasonable or sensible ? 2. A figure drawn to
scale makes it possible to measure the unknown parts and to
compare the results of such measurements with the computed
results. 3. An accurate check can often be secured with com
paratively little additional labor by computing one of the
quantities from two different formulas or by verifying a
known relation. For example, if the legs a, b of a right tri
angle have been computed by the formulas a = c sin A and
b = c cos A, we may check by verifying the relation a 2 + b 2 = c 2 .
II, § 21]
THE RIGHT TRIANGLE
27
Example. A straight road is to be built from a point A to a point B
which is 5.92 miles east and 8.27 miles north of
A. What will be the direction of the road and
its length ?
5.92 , D 8.27
Formulas :
Therefore
tan A =
AB =
8 27 cos A
tan A = 0.716 and A = 35° 35',
cos ^ = 0.813 ^£ = 10.17.*
Check by a 2 + & 2 = c 2 .
From a table of squares (p. 107, see § 21)
(5.92) 2 = 35.05
(8.27) 2 = 68.39 (10.17) 2 = 103.4.
103.4
21. Use of Table of Squares. Square Roots. The table
of squares of numbers (p. 106) may be used to facilitate com
putation. In the example of the last article, we required the
square of 5.92. We find 5.9 on p. 107 in the lefthand column
and find the third digit 2 at the head of a certain column. At
the intersection of the line and column thus determined we
find the desired result (5.92) 2 = 35.05. The square of 8.27 is
found similarly at the intersection of the line corresponding
to 8.2 and the column headed 7. To find (10.17) 2 , we find the
line corresponding to 1.0 (the first two digits, neglecting the
decimal point) and find (1.01)* = 1.020 and (1.02) 2 = 1.040.
By interpolating, as explained in § 18, we find (1.017) 2 = 1.034.
Now shifting the decimal point one place in the "number"
requires a corresponding shift of two places in the square.
Hence, (10.17)* = 103.4.
The table can also be used to find the square root of a num
ber. Thus to find V2 we find, on working backwards in this
table, that 2 lies between 1.988 [=(1.41)*] and 2.016 [=(1.42)*].
By interpolation we then find V2 m 1.414, correct to four
significant places. [Tabular difference = 28 ; correction = *^
= 4 in the fourth place.]
♦The retention of four significant figures in AB is justified because the
number is so small at the left.
^
28 PLANE TRIGONOMETRY [II, § 21
EXERCISES
1. From an observing station 357 ft. above the water, the angle of
depression of a ship is 2° 15 f . Find the horizontal distance to the ship in
yards .
2l A projectile falls in a straight line making an angle of 25° with the
horizontal. Will it strike the top of a tree 24 meters high which is 72 meters
from the point where the projectile would strike the ground ?
3. At a point 372 ft. from the foot of a cliff surmounted by an observa \*~* T
ion tower the angle of elevation of the top of the tower is 51° 25', and of 2**}.^
the foot of the tower 31° 55'. Find the height of the cliff and of the
tower.
f*. How far from the foot of a flagpole 130 ft. high must an observer
stand so that the angle of elevation of the top of the pole will be 25° ?
5. GA is a horizontal line, T is a point vertically above i; 5a point
AG
vertically below A. The angle BG A in minutes is Find Z BG T
4000
in degrees and minutes, given GA = 10,340 meters ; AT = 416.4 meters.
6. It is desired to find the height of a wireless tower situated on the
top of a hill. The angle subtended by the tower at a point 250 ft. below"
the base of the tower and at a distance measured horizontally of 2830 ft.
from it is found to be 2° 42'. Find the height of the tower.
7. From a tower 428.3 ft. high the angles of depression of two objects
tuated in the same horizontal line with the base of the tower and on the
same side are 30° 22' and 47° 37'. Find the distance between them.
8. The summit of a mountain known to be 13,260 ft. high is seen at
an angle of elevation of 27° 12' from a camp located at an altitude of
6359 ft. Compute the airline distance from the camp to the summit of
the mountain.
9. Two towns A and B, of which B is 25 miles northeast of A, are to
be connected by a new road. 11 miles of the road is constructed from
A in the direction N. 21° E. ; what must be length and direction of the
remainder of the road, assuming it to be straight ?
22. Applications in Navigation. We shall confine ourselves
to problems interne sailing; i.e. we shall assume that the dis
tances considered are sufficiently small so that the curvature of
the earth may be neglected.
II, § 22]
THE RIGHT TRIANGLE
29
Definition. The course of a
ship is the direction in which she
is sailing. It is given either by
the points of a mariner's compass
(Fig. 21) as K E. by N. or in
degrees and minutes ■ measured
clockwise from the north. Observe
that a " point " on a mariner's
compass is 11° 15'. Hence for
example, the course of a ship
could be given either as N. E. by
N. or as 33° 45 ; . A course S. E. by S. is the same as a course
of 146° 15'.
The distance a ship travels on a given course is always given
Departure in nautical miles or knots. A knot is the length
of a minute of arc on the earth's equator. (The
earth's circumference is then 360 x 60 = 21,600
knots.) The horizontal component of the dis
tance is called the departure, the vertical com
ponent is called the difference in latitude. The
departure is usually given in miles (knots), the
difference in latitude in degrees and minutes.
Fig. 22
Example. A ship starts from a position in 22° 12' N. lati
tude, and sails 321 knots on a course of 31° 15'. Find the
difference in latitude, the departure, and the latitude of the
new position of the ship,
diff. in lat. = distance times cosine of course
= 321 cos 31° 15'
= 321 x 0.855 = 274' = 4° 34'.
departure = distance times sine of course
= 321 sin 31° 15'
= 321 x 0.519 = 167 knots.
Since the ship is sailing on a course which increases the lati
30 PLANE TRIGONOMETRY [II, § 22
tude, the latitude of the new position is 22° 12' f 4° 34' = 26°
46' N.
Knowing the difference in latitude and the departure, we are
able to calculate the new position of the ship, if the original
position is known. In the preceding example, we found the
latitude of the new position from the difference in latitude.
To find the difference in longitude from the departure is not
quite so simple. As the latitude increases, a given departure
implies an increasing difference in longitude. Only on the
equator is the departure of one nautical mile equivalent to a
difference in longitude of one minute.
The adjacent figure shows a departure AB in latitude <f>.
The difference in longitude (in minutes) corresponding to AB
is clearly the number of nautical miles in
CD. Now arcs AB and CD are proportional
to their radii PA and OC. Or,
CD = ° C ~ . AB = A**. (Why ?)
PA cos<j> v J J
In practice, it is customary to take for <f>
in the determination of difference in longi
tude the socalled middle latitude, i.e. the
latitude halfway between the original latitude and the final
latitude.
Thus in the preceding example, the original latitude was
22° 12' N, the final latitude was 26° 46' N. The middle lati
tude is therefore J (22° 12' + 26° 46') = 24° 29'. Hence
,pp , , , departure
difference in longitude = . . , ,, — , — ■, — =r
cosme or middle latitude
167 167 = lg4 , m 30 4 ,
Fig. 23
cos 24° 29' 0.910
The determination of the position of a ship from its course
and distance is known as dead reckoning. It is subject to con
siderable inaccuracy and must often in practice be checked by
II, § 22] THE RIGHT TRIANGLE 31
direct determination of position by observations on the sun
or stars.
EXERCISES
1. A ship sails N. E. by E. at the rate of 12 knots per hour. Find the
rate at which it is moving north.
2. A ship sails N. E. by N. a distance of 578 miles. Find its departure
and difference in latitude.
3. A ship sails on a course of 73° until its departure is 315 miles. Find
the actual distance sailed. Find also its difference in latitude.
4. A ship sails from latitude 47° \& N. 670 miles on a course N. W.
by N. Find the latitude arrived at.
5. A ship sails from latitude 30° 24' N. and after 25 hours reaches lati
tude 35° 26' N. Its course was N. N. W. Find the average speed of the
ship.
6. A vessel sails from lat. 24° 30' N., long. 30° 15 W., a distance of 692
miles on a course of 32° 20'. Find the latitude and longitude of its new
position.
7. A vessel sails from lat. 10° 30' S., long. 167° 20' W., a distance of
692 miles on a course of 152° 30 f . Find the latitude and longitude of its
new position.
CHAPTEE III
SIMPLE TRIGONOMETRIC RELATIONS
/2Z. Other Trigonometric Functions. The reciprocals of
' the sine, the cosine, and the tangent of any angle are called,
respectively, the cosecant, the secant, and the cotangent of
that angle. Thus,
cosecant = dlstance of P =  (provided y =#= 0).
ordinate of P y
, r. distance of P r , . , q , AN
secant = — — : =  (provided x^=0).
abscissa of P x
f\ nsoissji Or r^ ^v
cotangent = : — — —  (provided y ^= 0).
ordinate of P p
These functions are written esc 0, sec 0, ctn 0. From the
definitions follow directly the relations
esc 6= — , sec 8 = , ctn 6
sin ' cos 9 ' tan 8
or
esc • sin = 1, sec 6 • cos = 1, ctn • .tan = 1.
To the above functions may be added versed sine (written versin), the
co versed sine (written coversin), and the external secant (written exsec),
which are defined by the equations versin = 1 — cos 0, coversin =
1 — sin 0, and exsec = sec — 1. Of importance in navigation and service
able in other applications (see § 88) is the haversine (written hav)
which is defined to be equal to one half the versed sine ; i.e.
have = (1 — cos 0).
r 24. The Representation of the Functions by Lines. Con
sider an angle in each quadrant and about the origin draw
32
Ill, § 24] SIMPLE TRIGONOMETRIC RELATIONS 33
Fig. 24
a circle of unit radius. Let P(x, y) be the point where the
circle meets the terminal side of 6. Then
sin 6 = ¥=zy, cos 6 = ^ = x,
i.e. the sine is represented by the ordinate of P and the cosine
by the abscissa. Hence the sine and cosine have respectively
the same signs as the ordinate and abscissa of P.
If we draw a tangent to the circle at the point A where the
Fig. 25
circle meets the a^axis and let the terminal line of 9 meet this
tangent in Q, we have
tenO = ^Q = AQ, sec0 = ^2=OQ.
Note that when 6 = 90°, 270°, and in general 90 + n . 360°,
270° + n • 360°, where n is any integer, there is no length AQ
cut off on the tangent line and hence these angles have no
tangents.
If we draw a line tangent to the circle at the point B where
D
34
PLANE TRIGONOMETRY
[HI, § 24
the circle cuts the yaxis and let the terminal line of 6 cut
this tangent in B, we have
ctn0=z BK =B ^ and csc £ = OR = QR
Fig. 26
EXERCISES
1. From Fig. 24 prove sin 2 + cos 2 = 1.
2. From Fig. 25 prove 1 + tan 2 = sec 2 0.
I.' From Fig. 26 prove 1 + ctn 2 as csc 2 0.
It Colon.
A^
B*N
/\\
/ ^
1 eK
^
\^
25. Relations among the Trigonometric Functions. As
one might imagine, the six trigonometric functions sine, cosine,
tangent, cosecant, secant, cotangent are connected by certain
relations. We shall now find some of these relations.
From Fig. 9 (§ 10) it is seen that for all cases we have
(1) x 2 + y 2 = r 2 .
If we divide both sides of (1) by r 2 , we have
+
v
or
sin 2 6 + cos 2 6 = 1.
Dividing both sides of (1) by x 2 , we have
1 (by hypothesis r =£ 0) ;
1+ %=b ( if **°>
Therefore,
1 + tan 2 6 = sec 2 6.
Similarly dividing both sides of (1) by y 2 gives
or
r
ctn 2 6 + 1
+ ! = , (i*y*0);
r
csc 2 6.
Ill, § 26] SIMPLE TRIGONOMETRIC RELATIONS 35
Moreover, we have
y
tane = ^ = :=^°
x x cos 8
i r
and, similarly,
• cos 6
ctn6 = ^— S.
, sin 8
26. Identities. By means of the relations just proved
any expression containing trigonometric functions may be
put into a number of different forms. It is often of the
greatest importance to notice that two expressions, although
of a different form, are nevertheless identical in value. (How
was an " identity" defined in algebra ?)
The truth of an identity is usually established by reducing
both sides, either to the same expression, or to two expres
sions which we know to be identical. The following examples
will illustrate the methods used.
Example 1. Prove the relation sec 2 + esc 2 = sec 2 esc 2 0.
We may write the given equation in the form
+ ^— = sec 2 esc 2 0,
cos 2 sin 2
sin 2 + cos 2
cos 2 sin 2
1
= sec 2 esc 2 0,
= sec 2 esc 2 0,
which reduces to
cos 2 sin 2
sec 2 esc 2 = sec 2 esc 2 0.
Since this is an identity, it follows, by retracing the steps, that the
given equality is identically true.
Both members of the given equality are undefined for the angles 0°, 90°,
180°, 270°, 360°, or any multiples of these angles.
cos 2
Example 2. Prove the identity 1 4 sin —
J 1  sin
Since cos 2 = 1 — sin 2 0, we may write the given equation in the form
1 + sin = 1 "" S1 " 2 9 or 1 + sin = 1 + sin 0.
1  sin
36 PLANE TRIGONOMETRY [III, § 26
As in Example 1, this shows that the given equality is identically true.
The righthand member has no meaning when sin = 1 , while the left
hand member is defined for all angles. We have, therefore, proved that
the two members are equal except for the angle 90° or (4 nf 1)90°, where
n is any integer.
The formulas of § 25 may be used to solve examples of the
type given in § 16.
Example 3. Given that sin = ft and that tan is negative, find the
values of the other trigonometric functions.
Since sin 2 + cos 2 = 1, it follows that cos = ± Jf , but since tan is
negative, lies in the second quadrant and cos0 must be — . More
over, the relation tan = sin 0/cos gives tan = — ft. The reciprocals
of these functions give sec = — , esc = y, ctn — — *g.
EXERCISES
1. Define secant of an angle ; cosecant ; cotangent.
2. Are there any angles for which the secant is undefined ? If so,
what are the angles ? Answer the same question for cosecant and co
tangent.
3. Define versed sine ; co versed sine ; haversine.
4. Complete the following formulas :
sin 2 6 + cos 2 = ? 1 + tan 2 = ? 1 + ctn 2 = ? tan = ?
Do these formulas hold for all angles ?
5. In what quadrants is the secant positive ? negative ? the cosecant
positive ? negative ? cotangent positive ? negative ?
6. Is there an angle whose tangent is positive and whose cotangent is
negative ?
7. In what quadrant is an angle situated if we know that
(a) its sine is positive and its cotangent is negative ?
(b) its tangent is negative and its secant is positive ?
(c) its cotangent is positive and its cosecant is negative ?
— ' — *«••«£• Express sin 2 + cos so that it shall contain no trigonometric
sA function except cos 0.
9. Transform (1 + ctn 2 0)csc so that it shall contain only sin 0.
10. Which of the trigonometric functions are never less than one in
absolute value ?
11. For what angles is the following equation true : tan = ctn ?
"^^wU 12. How many degrees are there in when ctn = 1? ctn — — 1 ?
sec = V2 ? esc = V£ ?
( H <^c3?
Ill, § 27] SIMPLE TRIGONOMETRIC RELATIONS 37
13. Determine from a figure the values of the secant, cosecant, and
cotangent of 30°, 150°, 210°, 330°.
14. Determine from a figure the values of the secant, cosecant, and
cotangent of 45°, 135°, 225°, 315°.
15. Determine from a figure the values of the sine, cosine, tangent,
secant, cosecant, and cotangent of 60°, 120°, 240°, 300°.
16. Find from the following equations.
(a) sin0=£. (i) tan0= — 1.
(6) sin =  \. (j) ctn =  1.
(c) cos = \. (k) tan = 1.
(d) cos =  £. (I) ctn = 1.
(e) sec = 2. (m) tan 2 = 3.
(/) sec =  2. (n) sin = 0.
(gr) esc = 2. (b) cos = 0.
(h) esc =• 2. O) tan = 0.
Prove the following identities and state for each the exceptional values
of thjt variables, if any, for which one or both members are undefined :
cos
tan =
sin0.
sin
ctn =
COS0.
14
sin0
COS0
tor* ( H/»*— • C^r
cos 1 — sin
sin 2 — cos 2 = 2 sin 2 01.
(1 — sin 2 0)csc 2 = ctn 2 0.
tan + ctn = sec esc 0.
[x sin + y cos 0] 2 4 [x cos — y sin 0] 2 = x 2 f y 2 .
2^ =C os0.
tan + ctn
1 — ctn4 = 2 esc 2  esc 4 0.
26. tan 2  sin 2 = tan 2 sin? 0.
27. 2(1 + sin 0) (1 + cos 0) = (1 + sin + cos 0) 2
28. sin 6 + cos« 0=13 sin 2 cos 2
esc esc
1
27. The Trigonometric Functions of 90° 0. Figure 21
represents angles 6 antf 90° — 0, when is in each of the four
X03
(vr ^Av^
/
(ooo V y\>^~
uuSe*^
38
PLANE TRIGONOMETRY
[III, § 27
quadrants. Let OP be the terminal line of and OP' the
terminal line of 90°  0. Take OP' = OP and let (x, y) be
Fig. 27
the coordinates of P and (x', y') the coordinates of P\
in all four figures we have
x ' = y> y f = x > r' = r.
Then
Hence
sin(9O°0) = ^ = :
cos
Also,
cos (90°  0) =  = 2 = sin 0,
r r
tan (90° 6) = ^ = =ctn0.
x' y
esc (90° — 0)=sec0,
sec (90° 0)= esc 0,
ctn (90° 0)= tan 0.
Definition. The sine and cosine, the tangent and cotangent,
the secant and cosecant, are called cofunctions of each other.
The above results may be stated as follows : Any function
of an angle is equal to the corresponding cofunction of the com
plementary angle.*
28. The Trigonometric Functions of 180° — 6. By draw
ing figures as in § 27, the following relations may be proved :
sin (180°  6) = sin 0, esc (180°  6) = esc 0,
cos (180°  0) =  cos 0, sec (180°  0) =  sec 0,
tan (180°  0) =  tan 0, ctn (180°  0) =  ctn 0.
The proof is left as an exercise.
* Two angles are said to be complementary if their sum is 90°, regardless
of the size of the angles.
Ill, § 29] SIMPLE TRIGONOMETRIC RELATIONS 39
29. The result of § 27 shows why it is possible to arrange
the tables of the trigonometric functions with angles from 0°
to 45° at the top of the pages and angles from 45° to 90° at
the bottom of the pages. For example, since sin (90° — 0) = cos 0,
the entry for cos will serve equally well for sin (90° — 6).
As particular instances we may note sin 67° = cos 23°, tan 67°
= ctn 23°, cos 67° = sin 23°. Verify these from the table.
The result of § 28 enables us to find the values of the func
tions of an obtuse angle from tables that give the values only
for acute angles. It will be noted that § 28 says that any
function of an obtuse angle is in absolute value equal to the same
function of its supplementary angle but may differ from it in
sign.
Thus to find tan 137° we know that it is in absolute value
the same as tan (180°  137°) = tan 43° = 0.9325. But tan 137°
is negative. Hence
tan 137° =  0.9325.
Similarly, sin 137°= 0.6820.
cos 137° =  0.7314.
EXERCISES
Find the values of the following :
tan 146°, sin 136°, cos 173°, tan 100°, cos 96°, sin 138°,
tan 98°, sin 145°, cos 168°, cos 138°, tan 173°, cos 157°.
CHAPTER IV
OBLIQUE TRIANGLES
30. Law of Sines. Consider any triangle ABC with the
altitude CD drawn from the vertex C (Fig. 28).
In all cases we have sin A
Therefore, dividing, we obtain
sin A a a
=  , or
sin B b sin A
(i)
(2)
sin B
If the perpendicular were dropped from B, the same argu
ment would give a/sin A = c/sin C. Hence, we have
a b c
sin A sin B sin C
This law is known as the law of sines and may be stated as
follows : Any two sides of a triangle are proportional to the
sines of the angles opposite these sides.
31. Law of Cosines. Consider any triangle ABC with the
altitude CD drawn from the vertex C (Fig. 29).
In Fig. 29 a
AD = b cos A ; CD = b sin A ; DB = c — b cos A.
In Fig. 29 b
AD = — b cos A ; CD = b sin A ; DB = c — b cos A.
In both figures
a2 = DB 2 f CZ) 2 .
40
d
IV, § 32]
Therefore
OBLIQUE TRIANGLES
41
a 1 = c 2  2 be cos A + b 2 cos 2 A + b 2 sin 2 A
= c 2 — 2bc cos ^ + (cos 2 A + sin 2 ^1)6 2 ,
o c
whence
a 2 _ tf. + C 2 _ 2 be cos i4.
The result holds also when A is a right angle. Why ?
Similarly it may be shown that
b 2 = c 2 + a 2 — 2 ca cos £,
c 2 = a 2 + b 2 — 2 a6 cos C.
Any one of these similar results is called the law of cosines.
It may be stated as follows :
Tlie square of any side of a triangle is equal to the sum of the
squares of the other two sides diminished by twice the product of
these two sides times the cosine of their included angle*
32. Solution of Triangles. To solve a triangle is to find
the parts not given, when certain parts are given. From
geometry we know that a triangle is in general determined
when three parts of the triangle, one of which is a side,
are given.f Eight triangles have already been solved
(§ 15), and we shall now make use of the laws of sines and
cosines to solve oblique triangles. The methods employed
will be illustrated by some examples. It will be found
advantageous to construct the triangle to scale, for by so doing
one can often detect errors which may have been made.
* Of what three theorems in elementary geometry is this the equivalent ?
t When two sides and an angle opposite one of them are given, the triangle
is not always determined. Why ?
42 PLANE TRIGONOMETRY [IV, § 33
33. Illustrative Examples.
Example 1. Solve the triangle AB C, given
= 276 A = 30° 20', B = 60° 45', a = 276.
Solution :
C = 180°  (A+ B) = 180°  91° 5' = 88° 55';
: _ a sin B _ 270 sin 00° 45' = (270) (0*8725) = 476 9 .
sin^l ' sin 30° 20' 0.5050
also
c  ^_ sill _^  276 sin 88° 55' _ ( 276 ) (0.9998) __ 546 4
sin A sin 30° 20' 0.5050
"Check : It is left as an exercise to show that for these values we have
c 2 = a 2 + b 2 — 2 ab cos C.
Example 2. Solve the triangle ABC, given
A = 30°, b = 10, a = 6.
{? Constructing the triangle ABC, we see that
two triangles AB X C and AB 2 C answer the descrip
* tion since b > a > altitude CD.
Solution : Now
***! = *, or sin B, = ^^ =0.833,
sin A a a
whence B\ = 56°. 5.
But
B 2 = 180°  B x = 180°  56°.5 = 123°.5,
and
Ci = 180° {A + 50= 180°  86°.5 = 93°. 5,
C 2 = 180°  {A + ft) = 180°  153°. 5 = 26 u .5.
Now
C2 _ sin C 2 or C2 _ a sin C 2 _ (6) (0.446) _ g 35
a sin ^1 ' sin 4 0.500
Also
Ci = 8 inC 1 . or Ci= asinC 2 == (6)(0.998)_ 1198
a sin 4 ' sin J. 0.500
Check: Ci 2 = a 2 + & 2 — 2 ab cos 0i.
143.5 = 36 + 100 +(2) (6) (10) (0.061) = 143.3.
C2 2 = a 2 + ^2 _ 2 a& cos C 2 .
28.62 = 36 + 100(2)(6)(10)(0.895) = 28.60.
IV, § 33]
OBLIQUE TRIANGLES
43
Example 3. Solve the triangle ABC, given a = 10, 6 = 6, C = 40°.
Solution : c 2 = a 2 + 6 2 — 2 ab cos (7
= 100 + 36  (120) (0.766)= 44.08.
Therefore c = 6.64. Now
sin ^ = asinC = (10)(0.643) =
c 6.64 '
i.e. A = 104°. 5. Likewise,
sing = 6sinC = (6X0.643) =
c 6.64 '
Check : A + B + C = 180°.0.
Example 4. Solve the triangle ABC when
C a = 7, 6 = 3, c = 5.
From the law of cosines,
&2 i C 2 _ a 2 i
COSA= 26c =, = 0.800,
cos B = ?l±^l»! = 15 = 0.928,
2 ac 14
co S C = ?l+^li? = 11 = 0.786.
2 06 14
i.e.
B
= 35°
.5.
s*
= S
JS*
a =
Fig.
7
33
Therefore
.4 = 120°, Z* = 21°.8, C = 38°.2.
Check : A + B + C = 180°.0
EXERCISES
\£) Solve the triangle ABC, given
/4 (a) ^1 = a0°, B = 70°,
a = 100 ;
(6) A = 40°, B = 70°,
c = 110;
xj\c) A = 45°.5, <7 = 68°.5,
6 = 40;
\d) B=60°.5, C = 44°20',
c = 20;
e)Va = 30, & = 54, C = 50° ;
^ a = 10,
6 = 12, c = 14 ;
2Q 6 = 8, a = 10, C = 60° ;
Jtf) a = 21,
6 = 24, c = 28.
2. Determine the number of solutions of the triangle ABC when
(a) A = 30°, 6 = 100, a = 70
(6) 4 = 30°, 6 = 100, a = 100
(c) 21 = 30°, 6 = 100, a = 50
(d) 4 = 30°, 6 = 100, a = 40
(e) A = 30°, 6 = 100, a = 120 ;
(/) J. = 106°, 6 = 120, a = 16 ;
(gr) 4= 90°, 6= 15, a = 14.
44
PLANE TRIGONOMETRY
[IV, 33
3. Solve the triangle ABC when
(a) A = 37° 20', a = 20, 6 = 26 ; (c) 4 = 30°, a = 22, 6 = 34.
(6; ^L = 37° 20', a = 40, 6 = 26;
( 4/^ h order to find the distance from a point A to a point B, a line
4C and the angles CAB and .A (72? were measured and found to be
300 yd., 60° 30', 5.6° 10' respectively. Find the distance AB.
5. In a parallelogram one side is 40 and one diagonal 90. The angle
between the diagonals (opposite the side 40) is 25°. Find the length of
the other diagonal and the other side. How many solutions ?
6. Two observers 4 miles apart, facing each other, find that the angles
of elevation of a balloon in the same vertical plane with themselves are
60° and 40° respectively. Find the distance from the balloon to each
observer and the height of the balloon.
7. Two stakes A and B are on opposite sides of a stream ; a third
stake C is set 100 feet from A, and the angles A C B and CAB are observed
to be 40° and 110°, respectively. How far is it from A to B ?
8. The angle between the directions of two forces is 60°. One force
is 10 pounds and the resultant of the two forces is 15 pounds. Find the
other force.*
9. Eesolve a force of 90 pounds into two equal components whose
directions make an angle of 60° with each other.
10. An object B is wholly inaccessible and invisible from a certain
point A. However, two points C and D on a line with A may be found
such that from these points B is visible. If it is found that CD = 300 feet,
AC = 120 feet, angle DCB = 70°, angle CDB  50°, find the length AB.
11. Given a, 6, A, in the triangle ABC. Show that the number of
possible solutions are as follows :
A<90°
f a < b sin A no solution,
I b sin A < a < b two solutions,
a>b
one solution.
 a = b sin A j
^^90°
(a_6 no solution,
a > b one solution.
12. The diagonals of a parallelogram are 14 and 16 and form an angle
of 50°. Find the length of the sides.
* It is shown in physics that if the line segments AB
and AC represent in magnitude and direction two
forces acting at a point A, then the diagonal AD of the
parallelogram ABCD represents both in magnitude and
direction the resultant of the two given forces.
IV, § 34]
OBLIQUE TRIANGLES
45
13. Resolve a force of magnitude 150 into two components of 100 and
80 and find the angle between these components.
14. It is sometimes desirable in surveying to extend a line such as AB
in the adjoining figure. Show that this can be done by means of the
broken line ABCDE. What measurements are necessary ?
15. Three circles of radii 2, 6, 5 are mutually tangent. Find the angles
between their lines of centers.
16. In order to find the distance between two objects A and B on op
posite sides of a house, a station C was chosen, and the distances CA
= 500 ft., CB = 200 ft., together with the angle ACB = 65° 30', were
measured. Find the distance from A to B.
17. The sides of a field are 10, 8, and 12
rods respectively. Find the angle opposite the
longer side.
18. From a tower 80 feet high, two objects,
A and B, in the plane of the base are found to
have angles of depression of 13° and 10° respec
tively ; thejiorizontal angle subtended by A and B at the foot C of the
tower jedBP. Find the distance from A to B.
Areas of Oblique Triangles.
When tivo sides andjjie included angle are given.
noting the area byQfjJire have from geometry
8 = i ch,
but h = b sin A ; therefore
(1) S = ±cbsmA.
Likewise,
S = i ab sin C and S = \ac sin B.
Fig. &i
2. When a side and two adjacent angles are given.
Suppose the side a and the adjacent angles B and C to be
given. We have just seen that 8 = \ ac sin B. But from the
law of sines we have
a sin C
sin A
46 PLANE TRIGONOMETRY [IV, § 34
Therefore
Q_ a 2 ' sin B ♦ sin C
2 sin^t
But sin A = sin [180°  (B + C)] = sin (5 + C). Therefore
q _ a 2 sin jB sin (7
^~ ~2 sin (B+C)'
j 3. jWhen the three sides are given.
^*"W e have seen that S = \ be sin A. Squaring both sides of
this formula and transforming, we have
£2 = 2_1 sin 2 ^l = — (lcos 2 ^l)
4 4
= 1(1 + 003.4). (1 cos^);
whence, by the law of cosines,
8*wmWl b * + c2  a2 \ bcf 1 fr 2 + c 2  a 2 ^
2\ 26c y 2^ 26c J
^ 2&c + & 2 + c 2 a 2 2 5c  b 2  c 2 + a 2
4 ' 4
_6+_c_ja 5fc — a a—b + c ( a+J^c>
~ 2 * 2 ' 2 * " 2
which may be written in the form
S 2 = s(sa)(sb)(sc),
where 2s = a + 6 + c. Therefore,
(2) S = Vs(s a)(s b) (sc).
f 35^ The Radius of the Inscribed Circle. If r is the radius
of£ne inscribed circle, we have from elementary geometry,
since s is half the perimeter of the triangle, S = rs ; equating
this value of 8 to that found in equation (2) of the last article
and then solving for r, we get,
v
(s — a)(s — b)(s — c)
s
IV, § 36]
OBLIQUE TRIANGLES
47
EXERCISES
Find the area of the triangle ABC, given
\> 1, a = 25, b = 31.4, C = 80° 25'. 4. a = 10, b = 7, C = 60°.
2 & = 24, c = 34.3, J. = 60° 25'. N» 5. a = 10, b = 12, C = 60°.
3. a = 37, 6 = 13, C = 40°. ^ 6. a = 10, 6 = 12, C = 8\
7. Find the area of a parallelogram in terms of two adjacent sides
and the included angle.
8. The base of an isosceles triangle is 20 ft. and the area is 100/V3
sq. ft. Find the angles of the triangle. Ans. 30°, 30°, 120°.
\j 9. Find the radius of the inscribed circle of the triangle whose sides
are 12, 10, 8.
10. How many acres are there in a triangular field having one of its
sides 50 rods in length and the two adjacent angles, respectively, 70°
and 60° ?
and 60° \
3,1
next
The Law of Tangents.
chapter the formulas in this
and the next article will be
needed.
Let CD be the bisector of
the angle G of the A ABC.
Through A draw a line II DC,
meeting BC produced in E.
Then CE = b. Why ? From
A draw a line q X DC meeting
CB in F. At F draw a line r J_ AF meeting AB in G.
AE=p.
Now AACF is isosceles. Why? The angle ACE = ZA
+ /.B and the bisector of Z.ACE is _L CD. Hence Z CAF
= Z CFA = ±Z(A + B). Moreover Z BAF= ZA±Z(A
+ B) = ±Z(AB).
Let
Now
tan
A + B
and tan
tan
tan
A + B
AB
48 PLANE TRIGONOMETRY [IV, § 36
But £ = !f = « + *. Why?
tan
Hence
tan
a,
. Angles of a Triangle in Terms of the Sides. Con
f struct the inscribed circle of the triangle
and. denote its radius by r.. If the perim
eter a + 6 + c = 2s, then (Fig. 36)
AE = AF=s a.
BD = BF=sb.
CD=CE = sc. .
_ i ti r . ■ , > r
Then tan i ^4 = , tan \ B — , tan I C =
s — a j s — b
where, from § 35, rrUr^L /\ .
= J (saXsb)(sc) _,
A F tc £+£>/* VA A >S^i> + <WJ> M
38. Solution of Triangles by Means of the Haversine.
The haversine may be used advantageously in the solution of triangles,
(1) when two sides and the included angle are given ; (2) when the
three sides are given. The law of cosines gives
2havJ. = 1  cos^l = 1  &2 + c2  a2
2 6c
_ q2_(fr _ c )2
2 be
or 4 6chav A = a 2 — (b — c) 2 .
1. If 6, c and A are given we may find a from the formula
(1) a 2 =(bc) 2 + Ibch&vA.
Similar formulas give b 2 or c 2 in terms of a, c, .B and a, 6, C respectively.
2. If a, 6, c are given, we may find A from the formula
(2) hav A = *,<» «)' = .('WQ •
w 4 be 6c
Similar formulas will give B and 0,
IV, § 38] OBLIQUE TRIANGLES 49
Example 1. Given A = 94°
23'.4, b = 55.12, c = 39.90. To find .
By formula (1) above :
6 = 55.12
be = 2199
c = 39.90
hav 94° 23'.4 = 0.0446
(6c) = 15.22
be hav A= 1184
(6c)2 = 231.6
4 6c hav J. = 4736
4 be hav A = 4736
a* = 4968
»
a = 70.49
Example 2. Given a = 4.51
, 6 = 6.13, c = 8.16. FindJL, B, C.
a2 = 2034 hav ^ = 1«^ = 0.0811 A= 33*05
(6c)2= 4.12 200.1
a 2_(6c)2= 16.22
be = 50.02
4 6c = 200.1
62= 37.58 hav 5 = 2426 =0.1648 B = 47° 54
( C _a)2= 13.32 147.21
62 _ ( C _ a )2 _ 24.26
ac= 36.80
4ac = 147.21
C 2= 66.59 hav C  63,97  0.5785 C  99° 02'
(6a)2= 2.62 110.58 Check; 18QO ,
C 2«(6_ a )2 = 63.97
ab= 27.646
4 a& = 110.58
EXERCISES
Solve the following triangles :
^ 1. a = 62.1, 6 = 32.7, c = 47.2.
^ 2. vl = 37°20', 6 = 2.4, c = 4.7.
N 3. B = 121° 32', a = 27.9, c = 35.8.
^ 4. a = 3.2, 6 = 5.7, c = 6.5.
5. C = 72°21'.4, a = 314.1, 6 = 427.3.
6. a = 346.1, 6 = 425.8, c = 562.3.
CHAPTER V
39. The Invention of Logarithms. The extensive numeri
cal computations required in business, in science, and in engi
neering were greatly simplified by the invention of logarithms
by John Napier, Baron of Merchiston (15501617). By means
of logarithms we are able to replace multiplication and division
by addition and subtraction, processes which we all realize are
more expeditious than the first two.
If we consider the successive integral powers of 2
a)
Exponent x
1
2
3
4
5
6
7
Result 2* . .
2
4
8
16
32
64
128
Exponent x
8
9
10
11
12
etc.
A. P.
Result 2* . .
256
512
1024
2048
4096
etc.
G. P.
we see that the results form a geometric progression (G. P.)
and the exponents an arithmetic progression (A. P.). We
know from elementary algebra that
and
x n
x n
Hence if we wish to multiply two numbers in our G. P. e.g.
4 x 8, we merely have to add the corresponding exponents 2.
and 3 and under the sum 5 find the desired product 32. Sim
ilarly, if we wish to divide e.g. 4096 by 128, we merely have to
subtract the exponent corresponding to 128, from that cor
50
V, § 39]
LOGARITHMS
51
responding to 4096 and under their difference 5 we find the
desired quotient 32.
To make the above plan at all useful it is evident that our
table must be expanded so as to contain more numbers. First
we can expand our table so that it will contain numbers less
than 2, by subtracting 1 successively from the numbers in the
A. P. and by dividing successively by 2 the numbers in the
G.P.
(2)
In the second place we may find new numbers by inserting
arithmetic means and geometric means. Thus, if we take the
following portion of the preceding table
5
4
3
2
1
1
2
3
4
5
6
7
0.03125
0.0625
0.125
0.25
0.5
1
2
4
8
16
32
04
128
2
 1
1
2
3
4
*

1
2
4
8
16
and insert between every two successive numbers of the upper
line their arithmetic, and between every two successive num
bers of the lower line their geometric mean, we obtain the
table
(3)
2
f
1
i
±
1
1
2
5
3
i
4
i
}V2
1
^V2
1
V2
2
2V2
4
4V2
8
8V2
16
If the radicals are expressed approximately as decimals, this
table takes the form
2.C
1.
5l.<
)0.5
0.5
1.0
1.5
2
2.5
3
3.5
4
0.25
0.35
0.50
0.72
1.00
1.41
2.00
2.83
4.00
5.66
8.00
11.31
16
52 PLANE TRIGONOMETRY [V, § 39
By continuing this process we can make any number appear
in the G. P. to as high a degree of approximation as we desire.
To prepare an extensive table, which gives values at small inter
vals, is quite laborious. However, it has been done, and we
have printed tables so complete that actual multiplication of
any two numbers can" be replaced by addition of two other
numbers. We shall soon learn how to use such tables.
40. Definition of the Logarithm. The logarithm of a
number JV to a base b (b > 0, =£ 1) is the exponent x of
the power to which the base b must be raised to produce the
number JV.
That is, if
&*= N,
then
x ^lo&AT.
These two equations are of the highest importance in all work
concerning logarithms. One should keep in mind the fact
that if either of them is given, the other may always be
inferred.
The numbers forming the A. P. in tables 1, 2, and 3 of § 39
are the logarithms of the corresponding numbers in the G. P.,
the base being 2. From table 3 we have 2* = 4 V2 which says
log 2 4V2 = .
EXERCISES
1. When 3 is the base what are the logarithms of 9, 27, 3, 1, 81, ,
2. Why cannot 1 be used as the base of a system of logarithms ?
3. When 10 is the base what are the logarithms of 1, 10, 100, 1000 ?
4. Find the values of x which will satisfy each of the following
equalities :
(a) log 3 27 = x. (d) log a a = x. (g) log 2 x = 6.
(6) \og x 3 = 1. (e) log a l=x. (h) log 32 z = .
(c) log, 5=. (/) log, b \ = x. ('J) logo.001 x = 2.
V, § 41] LOGARITHMS 53
5. Find the value of each of the following expressions :
(a) log 2 16. (c) loge^ (e) log 25 125.
(6) log 343 49. (d) log 2 Vl6. (/) log 2l fr.
41. The Three Fundamental Laws of Logarithms. From
the laws of exponents we derive the following fundamental
laws.
I. TJie logarithm of a product equals the sum of the logarithms
of its factors. Symbolically,
log 6 MN = log 6 M + log 6 N.
Proof. Let log 6 M = x, then b x = M. Let log 6 N= y, then
6 V = N. Hence we have MN = b x+y , or
log 6 MN ax m + y, i.e. log 6 MN = log,, M + log 6 N.
II. Tlie logarithm of a quotient equals the logarithm of the
dividend minus the logarithm of the divisor. Symbolically,
log 6 ^f= log 6 M  log & N.
N
Proof. Let log 6 M = x, then b* = M. Let log 6 N—y y then
b' J m N. Hence we have M/N= b*'", or
M M
^og b  = xy, i.e. \og b ^. = \og b M  \og b N.
III. The logarithm of the pth power of a number equals p
times the logarithm of the number. Symbolically
logfe M p = p log 6 M.
Proof. Let log 6 M = x, then b x = M. Raising both sides
to the pth power, we have b px = M v . Therefore
log 6 M p =px=p log, M.
Prom law III it follows that the logarithm of the real positive
nth root of a number is one nth of the logarithm of the number.
54 PLANE TRIGONOMETRY [V, § 41
2 EXERCISES
Given logi 2 = 0.3010, log 10 3 = 0.4771, logio 7 = 0.8451, find the
of each of the f ollowinggrxpressions :
(a) log w 6. (/) logi 5.
[Hint: logio 2x3= log 10 2 + logio 3.] [Hint: log 10 5 = log 10 y.J
(6) logio 21.0. (?) logio m
(c) logio 20.0. (h) logio Vl4.
(d) logio 0.03! (i) logio 49^_
(e) logio . (i) logio V24.7&.
2. Given the same three logarithms as in Ex. 1, find the value of each
of the following expressions : r> *
/„\ u„ 4 x 5 x 7 ,,x • ' 5 x 3 x 20 f * , 2058
(a) IogI °^2T^ (6) logl °^T^ N(c) loSl0 ^i
^(d) logio (2)*. (e) logic (3)8(5)«, (/) logio(2 3 )Q).
<5>
Logarithms to the Base 10. Logarithms to the base 10
are known as common or Briggian logarithms. Proceeding as
in § 39 we can show that 10  3010 = 2, i.e. log 10 2 = 0.3010. Let
ns multiply both members of the equation 10 03010 == 2 by 10, 10 2 ,
10 3 , etc. and notice the effect on the logarithm.
10 o.3oio = 2 log 10 2 = 0.3010
10 3010 = 20 log 10 20 = 1.3010
10 2.3oio = 200 log L0 200 = 2.3010.
It should be clear from this example that the decimal part of
the logarithm (called the mantissa) of a number greater than 1
depends only on the succession of figures composing the num
ber and not on the position of the decimal point, ^vhile the in
tegral part (called the characteristic) depends simply on the
position of the decimal point. Hence it is only necessary to
tabulate the mantissas, for the characteristics can be found by
inspection as the following considerations show.
Since
10° = 1, lO^lO, 10 2 = 100, 10 3 = 1000, 10 4 = 10,000, etc.
we have logj 1 = 0, log 10 10 = 1, log 10 100 = 2,
log,o 1000 = 3, log M 10,000 = 4, etc.
V, § 42] LOGARITHMS 55
It follows that a number with one digit (=f= 0) at the' left of the
decimal point has for its logarithm a number equal to 4 a
decimal ; a number with two digits at the left of its decimal
point has for its logarithm a number equal to 1 + a decimal ; a
number with three digits at the left of the decimal point has
for its logarithm a number equal to 2 + a decimal, etc. We
conclude, therefore, that the characteristic of the common loga
rithm of a number greater than 1 is one less than the number of
digits at the left of the decimal point.
Thus, logio 456.07 = 2.65903.
The case of a logarithm of a number less than 1 requires
special consideration. Taking the numerical example first con
sidered above, if log 10 2 =0.30103, we have log 10 0.2=0.301031.
Why? This is a negative number, as it should be (since the
logarithms of numbers less than 1 are all negative, if the
base is greater than 1). But, if we were to carry out this
subtraction and write log 10 0.2 = — 0.69897 (which would be
correct), it would change the mantissa, which is inconvenient.
Hence it is customary to write such a logarithm in the form
9.30103  10.
If there are n ciphers immediately following the decimal
point in a number less than 1, the characteristic is — n— 1.
For convenience, ifn< 10, we write this as (9 — n) — 10. TJiis
characteristic is written in two parts. The first part 9 — n is
ivritten at the left of the ma?itissa and the — 10 at the right.
In the sequel, unless the contrary is specifically stated, we
shall assume that all logarithms are to the base 10. We may
accordingly omit writing the base in the symbol log when there
is no danger of confusion. Thus, the equation log 2 = 0.30103
means log 10 2 = 0.30103.
To make practical use of logarithms in computation it is
necessary to have a conveniently arranged table from which
we can find (a) the logarithm of a given number and (b) the
number corresponding to a given logarithm. The general
./ <*
)
56 PLANE TRIGONOMETRY [V, § 42
principles governing the use of tables will be explained by the
following examples [Tables, pp. 110, 111].
Example 1. Find log 42.7.
The characteristic is 1. In the column headed N (p. 110) we find 42
and if we follow this row across to the column headed 7, we read 6304,
which is the desired mantissa. Hence log 42.7 = 1.6304.
Example 2. Find log 0.03273.
The characteristic is 8 — 10. The mantissa cannot be found in our
table, but we can obtain it by a process called interpolation. We shall
assume that to a small change in the number there corresponds a propor
tional change in the mantissa. Schematically we have
u ' ^' , Number Mantissa
difference = 10
. T3270 > 5145"
L3273 > ? 4 = difference
3280 — >» 5159 J
Our desired mantissa is 5145 + ^14 = 5149. Hence log 0.03273
= 8.5149  10.
Example 3. Find x when log x — 0.8485.
We cannot find this mantissa in our table, but we can find 8482 and
8488 which correspond to 7050 and 7060 respectively. Reversing the
process of example 2, we have schematically
Number Mantissa
"7050 < 84821 _"
Difference = 10 ? <— 8485 J 6 = difference
7060 < 8488
Hence the significant figures in our required number are 7050 f 1 • 10
= 7055. Since the characteristic is the required number is 7.055.
EXERCISES
/^) Find the logarithms of the following numbers from the table on
ppYllO, 111 : 482, 26.4, 6.857, 9001, 0.5932, 0.08628, 0.00038.
2. Find the numbers corresponding to the following logarithms :
2.W35, 0.3502, 7.9599  10, 9.5300  10, 3.6598, 1.0958.
43. Use of Logarithms in Computation. The way in
which logarithms may be used in computation will be suffi
ciently explained in the following examples. A few devices
often necessary or at least desirable will be introduced. The
V, § 43] LOGARITHMS 57
latter are usually selfexplanatory. Reference is made to
them here, in order that one may be sure to note them when
they arise. The use of logarithms in computation depends, of
course, on the fundamental properties derived in § 41.
Example 1. Find the value of 73.26 x 8.914 x 0.9214.
We find the logarithms of the factors, add them, and then find the
number corresponding to this logarithm. The work may be arranged as
follows :
Numbers
Logarithms
73.26
(►)
1.8649
8.914
(»
0.9501
0.9214
(»
9.9645  10
12.7795  10
Product = 601.9 Arts.
(«)
2.7795
Example 2. Find the value of 732.6 •
4 89.14.
Numbers
Logarithms
732.6
(*)
2.8649
89.14
(*0
1.9501
Quotient = 8.219 Ans.
(«)
0.9148
Example 3. Find the value of 89.14
= 732.6.
Numbers
Logarithms
89.14
c*o
11.9501  10
732.6
(>)
2.8649
Quotient = 0.1217 Ans.
«)
9.0852  10
Example 4. Find the value of *
x 21.63
Whenever an example involves several different operations on the
logarithms as in this case, it is desirable to make out a blank form. When
a blank form is used, all logarithms should be looked up first and entered
in their proper places. After this has been done, the necessary opera
tions (addition, subtraction, etc.) are performed. Such a procedure
saves time and minimizes the chance of error.
Form
Numbers Logarithms
763.3 (►)
21.63 (» ( + )
product
986.7 (» ().....
.... Ans. (<)
58
PLANE TRIGONOMETRY
[V, §43
Form Filled In
Numbers
Logarithms
763.2
(»
2.8826
21.63
(»)
1.3351
product
4.2177
986.7
(*)
2.9942
16.73 Ans.
«)
1.2235
Example 5. Find (1.357)5.
Numbers
Logarithms
1.357
(»
0.1326
(1.357)5 = 4.602
Ans.
(*)
0.6630
Example 6. Find the cube
root of 30.11.
Numbers
Logarithms
30.11
(»
1.4787
#30.11 =3.111
Ans.
(«)
0.4929
Example 7. Find the cube root of 0.08244.
Numbers Logarithms
0.08244 (>) 28.9161  30
#0.08244 = 0.4352 Ans. («<) 9.638710
6 D
EXERCISES
Compute the value of each of the following expressions using the table
on pp. 110, 111.
1. 34.96 x 4.65.
2. 518.7 x 9.02 x .0472.
3 0.5683 _
0.3216*
4 5.007 x 2.483
6.524 x 1.110*
5. (34.16 x .238)2.
6. 8.572 x 1.973 x (.8723)2.
K.#
8076 x 3.184
(2.012)5
O 10. a/ 2941 >< 17 
11.
'2173 x 18.75
#0.00732
#735
^12. (20.027)*
d/lS. 2 1( ».
■' i
e
648.8
'(21.4)1
/ 1379
>2791 "
/
v 14. Vio^.ioo 2 .
15. (0.02735)*.
die.
f A
#3275
(2.01)
y
i
V, § 44] LOGARITHMS 59
44. Cologarithms. Since — and M • —  are equivalent,
we may in a logarithmic computation, add the logarithm of
— instead of subtracting log N. The logarithm of — is
called the cologarithm of N. Therefore
colog N = log 1/N = log 1 — log N = — log N,
since log 1 is zero.
We write cologarithms, like logarithms, with positive man
tissas. Therefore the cologarithm is most easily found by sub
tracting the logarithm from zero, written in the form 10.0000
10.
Example. Find the colog 27.3.
10.0000  10
i log 27. 3= 1.4362
colog 27.3= 8.563810
The cologarithm can be written down immediately by subtracting the
last significant figure of the logarithm from 10 and each of the others
from 9. If the logarithm is positive the cologarithm is negative and
hence — 10 is affixed.
There is no gain in using cologarithms when we have a quotient of two
numbers. There is an advantage when either the numerator or denomi
nator contains two or more factors, for we can save an operation of addi
tion or subtraction. Let us solve Ex. 4, § 43, using cologarithms.
Example. Find the value of 763 ' 2 x 2L63 ■
986.7
Numbers Log
763.2 .>■ 2.8826
21.63 > 1.3351
986.7 > (colog) 7.0058  10
16.73 < 1.2235
EXERCISES
Compute the value of each of the following expressions, using cologa
rithms.
/?\ J 2.80 x 37.6 /"T\ J
97.63 x 876.5
2876 x3.4 x 2.987
60 PLANE TRIGONOMETRY [V, §44
3 5 5 V3275 ,
' 7 x 8 x 9 x 27.6 ^J (2.01)*(1.76)»
4. 312 • 6 1293 x 12 7 x 5
610,27 N ^(l + 2 V3)(760 + 8)'
MISCELLANEOUS EXERCISES
1. What objections are there to the use of a negative number as the
base of a system of logarithms ?
2. Show that a l °s a x = x.
3 . Write each of the following expressions as a single term :
'a) log x + log y — log z. QpS^P log x — 2 log y + 3 log z.
XcpS log a — log (x + y)  \ log (ex + tf)'+ log Vw + x.
4} Solve for x the following equations :
§2 log 2 £ + log 2 4 = 1. (c) 2 logio x  3 log 10 2 = 4.
log 3 x  3 log 3 2 = 4. (d) 3 log 2 x + 2 log 2 3 = 1, 
/5. How many digits are there in 2 35 ? 3 142 ? 3 12 x 2» ? ^g
y6. Which is the greater, (f£) 100 or 100 ?
/ 7> Find the value of each of the following expressions :
(a% log 6 35. ((py log 3 34. (g) log 7 245. (d) log 13 26. 
8. Prove that log b a • log a 6 = 1.
9. Prove that
log„ a; + V x2 ~  = 2 lo go [x + Vx 2  1].
« — Vx 2 — 1
10. The velocity v in feet per second of a body that has fallen s feet
is given by the formula v = V64.3s.
What is the velocity acquired by the body if it falls 45 ft. 7 in. ?
/ 11. Solve for x and ?/ the equations ; 2 X = 16v, x + 4 ?/ = 4.
►
m
CHAPTER VI
LOGARITHMIC COMPUTATION
46. Logarithmic Computation. In the last chapter a few
examples of the use of logarithms in computation were given
in connection with a fourplace table. Such a table suffices
for data and results accurate to four significant figures. When
greater accuracy is desired we use a five,' six, or sevenplace
table.
No subject is better adapted to illustrate the use of logarith
mic computation than the solution of triangles, which we shall
consider in some detail. Fiveplace tables and logarithmic
solutions ordinarily are used at the same time, since both tend
toward greater speed and accuracy.
46. Fiveplace Tables of Logarithms and Trigonometric
Functions. The use of a fiveplace table of logarithms differs
from that of a fourplace table in the general use of socalled
" interpolation tables " or " tables of proportional parts," to
facilitate interpolation. Since the use of such tables of pro
portional parts is fully explained in every good set of tables,
it is unnecessary to give such an explanation here. It will be
assumed that the student has made himself familiar with their
use.*
In" the logarithmic solution of a triangle we nearly always
need to find the logarithms of certain trigonometric functions.
For example, if the angles A and B and the side a are given,
we find the side b from the law of sines given in § 30,
, _ a sin B
♦^* sin A
* For this chapter, such a fiveplace tahle should be purchased. See, for
example, The Macmillan Tables, which contain all the tables mentioned
here with an explanation of their use.
61
62 PLANE TRIGONOMETRY [VI, § 46
To use logarithms we should then have to find log a, log (sin B)
and log (sin A). With only a table of natural functions and a
table of logarithms at our disposal, we should have to find first
sin A, and then log sin A. For example, if A = 36° 20', we
would find sin 36° 20' = 0.59248, and from this would find log
sin 36° 20' = log 0.59248 m 9.77268  10. This double use of
tables has been made unnecessary by the direct tabulation of the
logarithms of the trigonometric functions in terms of the angles.
Such tables are called tables of logarithmic sines, logarithmic
cosines, etc. Their use is explained in any good set of tables.
The following exercises are for the purpose of familiarizing
the student with the use of such tables.
J EXERCISES
V. Find the following logarithms : *
(a) log cos 27° 40'.5. (d) log ctn 86° 53'. 6.
(6) log tan 85° 20'.2. (e) log cos 87° 6'.2.
\}c) log sin 45° 40'. 7. (/) log cos 36° 53'. 3.
"■■k. Find A, when
(a) log sin A = 9.81632  10. (d) log sin A = 9.78332  10. •
(6) log cos A = 9.97970  10. (e) log ctn } A = 0.70352.
(c) log tan A = 0.45704. •(/) log tan \A = 9.94365  10.
VL Find Mf tan fl = 476  32 x 89  710 .
\ 87325
^ 4. Given a triangle ABC, in which ZA = 32°, Z B = 27°, a = 5.2, find
b by use of logarithms.
47. The Logarithmic Solution of Triangles. The effective
use of logarithms in numerical computation depends largely on
a proper arrangement of the work. In order to secure this,
the arrangement should be carefully planned beforehand by
constructing a blank form, which is afterwards filled in. More
over, a practical computation is not complete until its accuracy
has been checked. The blank form should provide also for a
good check. Most computers find it advantageous to arrange
* Fiveplace logarithms are properly used when angles are measured to the
nearest tenth of a minute. For accuracy to the nearest second, six places
should be used.
VI, § 48] LOGARITHMIC COMPUTATION 63
the work in two columns, the one at the left containing the
given numbers and the computed results, the one on the right
containing the logarithms of the numbers each in the same
horizontal line with its number. The work should be so
arranged that every number or logarithm that appears is
properly labeled ; for it often happens that the same number
or logarithm is used several times in the same computation and
it should be possible to locate it at a glance when it is wanted.
The solution of triangles may be conveniently classified
under four cases :
Case I. Given two angles and one side.
Case II. Given two sides and the angle opposite one of the
sides.
Case III. Given two sides and the included angle.
Case IV. Given the three sides.
In each case it is desirable (1) to draw a figure representing
the triangle to be solved with sufficient accuracy to serve as a
rough check on the results ; (2) to write out all the formulas
needed for the solution and the check ; (3) to prepare a blank
form for the logarithmic solution on the basis of these
formulas ; (4) to fill in the blank form and thus to complete
the solution.
We give a sample of a blank form under Case I ; the student
should prepare his own forms for the other cases.
48. Case I. Given Two Angles and One Side.
Example. Given: a=430.17, ^1=47° 13'.2, B=52° 29'.5. (Fig. 37.)
To find: C, 6, c.
Formulas :
C = 180°(A + B),
b=—2sinB,
sin A
sin C.
sin A
Check (§ 36): ^=± = tan $(<?*) .
^ J c + b tanJ(C+B) * Fig. 37
64
PLANE TRIGONOMETRY
[VI, § 48
The following is a convenient blank form for the logarithmic solu
tion. The sign (+) indicates that the numbers should be added ; the
sign (— ) indicates that the number should be subtracted from the one
just above it.
A =
( + )* =
A+ B =
C =
a =
sin A =
Numbers
179° 60'.0
Logarithms
sin
a/sin A
sin B = sin
b = . .
a/sin A
sin C
c
cb =
c+ b =
CB=. .
C+ B= . .
tan  ( C — B) = tan
tan \{C + B)= tan
()
») ( + )
H (+)
Check
■» ()
•) ()
(1)
(Logs (1) and (2)
. should be equal
. for check.)
"(2)
Filling in this blank form, we obtain the solution as follows.
Numbers
A= 47°13'.2
B= 52°29'.6
A+ B= 99°42'.8
179° 60'.0
Logarithms
0= 80°17'.2
a^= 430.17
sin A =sin47°13'.2
a/sin A
sin B = sin 52° 29'. 6
b = 464.94 Ans.
2.63364
() 9.86567  10
2.76797
( + ) 9.89943  10
2.66740
Check*
VI, § 49] LOGARITHMIC COMPUTATION 65
a/sin A 2.76797
sin C = sin 80° 17'. 2 (>) ( + ) 9.99373  10
c = 577.70 Ans. (<) 2.76170
Check
cb = 112.76 (>) 2.05215
c + b = 1042.64 (>) () 3.01813
9.03402  10
CB = 27°47'.6
C+£ = 132°46'.8
tan(C i*)=tanl3°53'.8 (>) 9.3934210
tan£(C + 5)= tan 66° 23'. 4 (>) () 0.35942
9.03400  10
EXERCISES
Solve *ud ulUWft the following triangles ABC :
. V. a = 372.5, ^4 = 25° 30', 5 = 47° 50'.
>* X c = 327.85, A = 110° 52'.9, 5 = 40° 31'.7. Ans. C = 28° 35'.4,
a = 640.11, 6 = 445.20.
3. a = 53.276, A = 108° 50'.0, C = 57° 13'. 2.
^ V b = 22.766, B = 141° 59M, C = 25° 12'.4.
5. b = 1000.0, B = 30° 30'.5, C = 50° 50'.8.
X, «' a = 257.7, J. = 47° 25', B = 32° 26'.
49. Case n. Given Two Sides and an Angle Opposite
One of Them.
If A, a, b are given, B may be determined from the relation
(1) AnB = bsmA 
a
If log sin B = 0, the triangle is a right triangle. Why ?
If log sin B > 0, the triangle is impossible. Why ?
If log sin B < 0, there are two possible values, B u B 2 of 5,
which are supplementary.
Hence there may be two solutions of the triangle. (See
Example.)
No confusion need arise from the various possibilities if the
corresponding figure is constructed and kept in mind.
It is desirable to go through the computation for log sin B
* A small discrepancy in the last figure need not cause concern. Why ?
66
PLANE TRIGONOMETRY
[VI, § 49
before making out the rest of the blank form, unless the data
obviously show what the conditions of the problem actually
Example L Given : A = 46° 22'.2, a = 1.4063, b = 2.1048. (Fig. 38.)
To find: B, C, c.
Formula : sin B = bsinA .
Fig. 38
Numbers Logarithms
6 = 2.1048 (>) 0.32321
sin A = sin 46° 22' .2 (>) ( + ) 9.85962 10
bsinA 0.18283
a =1.4063 (>) () 0.14808
sin B (<) 0.03475
Hence the triangle is impossible. Why ?
Example 2. Given : a = 73.221, b = 101.53, A = 40° 22'.3. (Fig. 39.)
To find : B, C, c.
Formula: sin£= &sin ^ .
Numbers Logarithms
b = 101.53 (>*) 2.00660
sin ^L= sin 40° 22'. 3 (>) ( + ) 9.81140  10
6sin^i
a = 73.221
sin i?
11.8180010
(>*) () 1.86464
9.95336  10
The triangle is therefore possible and
has two solutions (as the figure shows) .
We then proceed with the solution as
follows :
We find one value 2?i of B from
the value of log sin B. The other
value B 2 of B is then given by B 2 =
180°  B x .
VI, § 49] LOGARITHMIC COMPUTATION 67
Other formulas :
C= 180° (A + B).
a sin C
sin A
Check: ^^
c + b
_tanKCB) <
tan£(C + B)
Numbers
Logarithms
sin B
9.95336  10
i*i= 63° 55'. 2
179° 60\0
B 2 = 116° 4' .8
A + B x = 104° 17'.5
179° 60'.0
d= 75°42'.5
a
(>) 1.86464
sin .4
(►) () 9.8114010
a/ahiA
2.05324
sin d = sin 75° 42'. 5 (►) ( + ) 9.98634  10
d = 109.54 O) 2.03958
db= 8.01 (>) 0.90363
ci + 6 = 211.07 (>) () 2.32443
8.57920  10
C l B l = 11°47'.3
Ci + Bi = 139° 37 '.7
tan 4(Ci— JBi)= tan 5° 53'. 6 (►) 9.01377  10
tan K Ci + Bi) = tan 69° 48' . 8 (*►) 0.43455
8.57922  10
\ Check.
One solution of the triangle gives, therefore, B = 63° 55'. 2, C = 75° 42'. 5,
c = 109.54.
To obtain the second solution, we begin with B 2 = 116° 4'. 8. We find
C 2 from C 2 = 180°  (A + B 2 ); i.e. C 2 = 23° 32'. 9. The rest of the com
putation is similar to that above and is left as an exercise.
EXERCISES
1. Show that, given J., a, 6, if A is obtuse, or if J. is acute and a > 6,
there cannot be more than one solution.
Solve the following triangles and check the solutions :
J 2. a = 32.479, 6 = 40.176, A = 37° 25M.
68
PLANE TRIGONOMETRY
[VI, § 49
/:
3. 6 = 4168.2,
4. a = 2.4621,
5. a = 421.6,
6. a = 461.5,
3179.8,
4.1347,
532.7,
c = 121.2,
B = 51°21'A.
B = 101° 37'.3.
A = 49° 21 '.8.
C=22°31'.6.
7. Find the areas of the triangles in Exs. 25.
50. Case III. Given Two Sides and the Included Angle.
Example. Given: a=214.17, 6=356.21,
B C = 62° 21 '.4. (Fig. 40.)
/ N.
To find: A, B, c.
V ^v
Formulas :
V \
tan
(BA)= l L=Jtt^l(B + A);
p> \
B + A = 180°  O = 117° 38'.6
a sin C
C b = 356.Sl J.
Fig. 40
sin J.
Numbers
Logarithms
6  a = 142.04
c*o
2.15241
6 + a = 570.38
(»
() 2.75616
(6  a)/(b + a)
9.39625  10
tan (1? + A) = tan 58° 49'.3
(»
( + ) 0.21817
tan^(£ A)= tan 22° 22'. 2
(«)
9.61442  10
.. J.= 36°27'.l
.Ans.
2*= 81° 11'.5
J.W8.
a = 214.17
(— ►)
2.33076
sin^L = sin36°27'.l
(»
() 9.77389 10
a/sin .4
2.55687
sin C = sin 62° 21'. 4
(»
( + ) 9.9473610
c = 319.32 .4ns.
(«)
2.50423
Check by finding log (6/sin B).
I
SXERCI
SES f
Solve and check each of the following triangles :
1. a = 74.801, 6 = 37.502, C = 63°35'.5.
^ 2. a = 423.84, 6 = 350.11, G = 43° 14'.7.
s 3. 6 = 275, c = 315, A = 30° 30/.
4. a = 150.17, c = 251.09, B = 40°40'.2 ;
> 6. a = 0.25089, 6 = 0.30007, C = 42° 30' 20".
6. Find the areas of the triangles in Exs. 15.
VI, § 51] LOGARITHMIC COMPUTATION
69
51. Case
IV. Given the
Sides.
Example.
Given: a = 261.62,
6 = 322.42,
c = 291.48.
To find: A, B, C.
Formulas :
s = K«
+ b + c).
r _J(«
— a) — 6) (8 — c)
ryj
8
tan i A = r
, tani£ = ^,
s
a « — 6
Check : A + B + C = 180°.
Numbers
a = 261.62
6 = 322.42
c = 291.48
<
28 = 875.52
8 = 437.76
8 —
a = 176.14
8 
b = 115.34
8
 c = 146.28
tan \C
s — c
Logarithms
(►)
s = 437.76 (Check). *(
2.24586
2.06198
) ( + ) 2.16518
6.47302
) () 2.64124
3.83178
r
s 1 — a
tan  A  tan 25° 4'. 1
r
sb
tan£ Bz= tan35°32'.4
r =
s — c =
(«)
«)
(«)
1.91589
2.24586
9.67003  10
1.91589
2.06198
9.85391  10
191589
2.16518
9.75071  10
A= 50° 8'.2 Ans.
B= 71° 4'.8 ^?is.
C = 58° 46'.9 ^Ins.
179° 59'.9 (Check.)
"7Vys>
*By adding 8— a, 8 — 6, s
(A*)*
r^t (ft e}« t^. ^ fc A
(§37)
70 PLANE TRIGONOMETRY [VI, § 51
EXERCISES
Solve and check each of the following triangles :
VI. a as 2.4169, b = 3.2417, c = 4.6293.
*%!.<*= 21.637, & = 10.429, c = 14.221.
5. a as 528.62, . 6 = 499.82, c = 321.77.
4. a = 2179.1, 6 = 3467.0, c = 5061.8.
V« a = 0.1214, & = 0.0961, c = 0.1573.
6. Find the areas of the triangles in Exs. 15.
7. Find the areas of the inscribed circles of the triangles in Exs. 15.
OTHER LOGARITHMIC COMPUTATIONS
52. Interest and Annuities.
Simple Interest.
Let the principal be represented by P
the interest on $ 1 for one year by r
the number of years by n
the amount of P for n years by A n
Then the simple interest on P for a year is Pr
the amount of P for a year is P + Pr =P (14 r),
the simple interest on P for n years is Pnr
the amount of P for n years is A n =P(1 + nr).
Example. How long will it take $210, at 4% simple interest, to
amount to $ 298.20 ?
A n = P(l + nr) i.e. n = An ~ P .
Pr
Number Logarithm
A n  P = 88.20 > 1.9455
Pr= 8.40 ^ 0.9243
n = 10.5 «— 1.0212 10 yr. 6 mo. ^Ins.
Compound Interest.
Let the original principal be P
and the rate of interest r
Then the amount A] at the end of the first year is
A x = PhPr=:P(l\r),
VI, § 52] LOGARITHMIC COMPUTATION 71/
the amount A 2 at the end of the second year is
A 2 = A 1 (l + r) = P(l + ry,
the amount at the end of n years is
4,«J»(l+r)".
If the interest is compounded semiannually, A n — pf 1 + M ,
1+) , if q times a year^l n =P( 1 +  j ■
Since P in n years will amount to A H , it is evident that P at
the present time may be considered as equivalent in value to
A due at the end of n years. Hence P is called the present
worth of a given future sum A. Since
A n = P(l + r)% P= A n (1 + r)"\
Example. In how many years will one dollar double itself at 4 % in
terest compounded annually ?
A n = P(\ + r)  or log ^ = nlog(l + r).
. n = logAlogP
log (1 + r)
Hence n = log2  log 1 = 0,3010 = 17 . 7 .
log (1.04) 0.0170
17 yr. 9 mo. Ans.
Annuities. An annuity is a fixed sum of money payable
at equal intervals of time.
To find the present worth of an annuity of A dollars pay
able annually for n years, beginning one year hence, the rate
of interest being r and the number of years n.
Since the present worth of the first payment is A (1 + r) _1 ,
of the second A(l f r) 2 , etc., the present worth of the whole
is
P=^[(l + r)i+(lf r)*+ . +(l + r)*].
The quantity in the brackets is a G. P. whose ratio is (1 + r)~K
Summing, we have
l(l + r)i r\_ {1 + ryj
72 PLANE TRIGONOMETRY [VI, § 52
If the annuity is perpetual, i.e. n is infinite, the formula for
A
present worth becomes P — — •
Example. What should be paid for an annuity of $ 100 payable an
nually for 20 years, money being worth 4 % per annum ?
p=Mh LLl.
0.04 L (1.04) 20 J
20 = 2.188.
Therefore P= — fl L1 =2500 f U^§1 =$1358, approximately.
0.04 L 2.188 J L2.188J ' FF J
(1.04)
By logarithms (1 .04) 20  2. 188.
53. Projectiles. Logarithms are used extensively in ballis
tic computations. [Ballistics is the science of the motion of
a projectile.] The following is a very simple example of the
type of problem considered.
The time of flight of a projectile (in vacuum) is given by
the formula T=\ * where X is the horizontal range
* 9
in feet, <f> is the angle of departure, and g is the acceleration
due to gravity in feet per second per second \_g — 32.2]. If it
is known that the range is 3000 yd. and that the angle of de
parture is 30° 20', find the time of flight.
T /2Xtan<£
" X 9
Numbers
Logarithms
21= 18000
~*
4.2553
tan 30° 20'
*
9.7673  10
4.0226
32.2
"*
1.5079.
2)2.5147
18.09
<—
1.2574 T = 18.09 seconds.
Ans.
EXERCISES
1. Find the amount of $ 500 in 10 years at 4 per cent compound inter
est, compounded semiannually.
2. In how many years will a sum of money double itself at 5 per cent
interest compounded annually ? semiannually ?
VI, § 54] LOGARITHMIC COMPUTATION 73
3. A thermometer bulb at a temperature of 20° C. is exposed to the air
for 15 seconds, in which time the temperature drops 4 degrees. If the
law of cooling is given by the formula = doe 61 , where 6 is the final tem
perature, #o the initial temperature, e the natural base of logarithms, and
t the time in seconds, find the value of b.
4. The stretch s of a brass wire when a weight m is hung at its free
end is given by the formula j
8 = — — ,
where m is the weight applied in grams, g = 980, I is the length of the
wire in centimeters, r is the radius of the wire in centimeters, and fc is a
constant. If m = 844.9 grams, I = 200.9 centimeters, r = 0.30 centi
meter when s = 0.056, find k.
5. The crushing weight P in pounds of a wroughtiron column is given
by the formula ,73.55
P= 299,600^—,
p
where d is the diameter in inches and I is the length in feet. What weight
will crush a wroughtiron column 10 feet long and 2.7 inches in diameter?
6. The number n of vibrations per second made by a stretched string
is given by the relation 2 rzr
n = 2TV^r'
where I is the length of the string in centimeters, M is the weight in
grams that stretches the string, m the weight in grams of one centimeter
of the string, and g = 980. Find n when M = 5467.9 grams, I = 78.5
centimeters, m = 0.0065 gram.
7. The time t of oscillation of a pendulum of length I centimeters is
given by the formula ,— —
>(980
Find the time of oscillation of a pendulum 73.27 centimeters in length.
8. The weight w in grams of a cubic meter of aqueous vapor saturated
at 17° C. is given by the formula
= 1293 x 12.7 x 5
(1 + ^X760x8)*
Compute w.
54. The Logarithmic Scale. An arithmetic scale in which the
segments from the origin are proportional to the logarithms of 1, 2, 3, etc.,
is called a logarithmic scale. Such a scale is given in Fig. 42.
i I JIIIJ1
Fig. 42
74
PLANE TRIGONOMETRY
[VI, § 55
55. The Slide Rule. The slide rule consists of a rule along the
center of which a slip of the same material slides in a groove. Along the
Fig. 43
upper edge of the groove are engraved two logarithmic scales, A and B,
that are identical. Along the lower edge are also two identical logarithmic
scales, and D, in which the unit is twice that in scales A and B. Since
the segments represent the logarithms of the numbers found in the scale,
the operation of adding the segments is equivalent to multiplying the
f
1 2 £
*
1
4 I
) 6
7 i
5 9 1
2
A 1 ! 1 1 Ml ii i I I I
!
M l 1 1 mil ilmlilililililililili
iiilii
1,1,1 1
J EH
III
IIJJ 11,1,1 ,1,
jTI'II
WMV,
\
r oL
1
II
IK
II
II III
But
1
JIIIIJII
\
B r
\
!' ! i
\
D l
2
3^ 4
5 6 7 8 9
)
rl ■ ■
2
3
/
I C
1 1 ll Hill
llllll II
M'lll
Ii IjlJ ll
tiin
TtT&U
n, INI INI IIIIIHIIHI HI
nil
I If
MM 1
J.l
1.1 .
1J ftf4
.njijl
1
2
3
4
Fig. 44
corresponding numbers. Thus in Fig. 44 the point marked 1 on scale B
is set opposite the point marked 2.5 on scale A. The point marked 4 on
scale B will be opposite the point marked 10 on scale A, i.e. 2.5 x 4 = 10.
Similarly we read 2.5 x 3.2 = 8, 2.5 x 2.5 = 6.25. Other multiplications
can be performed in an analogous manner.
Division'can be performed by reversing the operation. Thus in Fig. 44
every number of scale B is the result of dividing the number above it by
2.5. Thus we read 7.2 ~ 2.5 = 2.9 approximately.
Since scales G and D are twice as large as scales A and B, it follows
that the numbers in these scales are the square roots of the numbers
opposite to them in scales A and B. Conversely the numbers on scales
A and B are the squares of the numbers opposite them on scales C and
D. Moreover the scales C and D can be used for multiplying and divid
ing, but the range of numbers is not so large.
For a more complete discussion of the use of a slide rule consult the
book of instructions published by any of the manufacturers of slide rules,
where also exercises will be found for practice.
CHAPTEK VII
TRIGONOMETRIC RELATIONS
56. Radian Measure. In certain kinds of work it is more
convenient in measuring angles to use, instead of the degree,
a unit called the radian. A radian is defined as the angle at
the center of a circle whose subtended arc is equal in length
to the radius of the circle (Fig. 45). Therefore, if an angle $
at the center of a circle of radius r units subtends an arc of
s units, the measure of 6 in radians is
r
Since the length of the whole circle is 2 nr, it follows that
— = 2tt radians = 360°,
r
or
(2) it radians = 180°.
Therefore,
180°
TT
1 radian = = 57° 17' 45" (approximately). FlG 45
It is important to note that the radian * as defined is a con
stant angle, i.e. it is the same for all circles, and can therefore
be used as a unit of measure.
From relation (2) it follows that to convert radians into
degrees it is only necessary to multiply the number of radians
by 180/7T, wliile to convert degrees into radians we multiply
the number of degrees by tt/180. Thus 45° is tt/4 radians ;
7r/2 radians is 90°.
* The symbol r is often used to denote radians. Thus 2 r stands for 2
radians, ir r for tt radians, etc. When the angle is expressed in terms of it (the
radian being the unit), it is customary to omit r . Thus, when we refer to an
angle it, we mean an angle of it radians. When the word radian is omitted,
it should be mentally supplied in order to avoid the error of supposing ir
means 180. Here, as in geometry, t = 3.14159. . . .
75
76 PLANE TRIGONOMETRY [VII, § 57
57. The Length of Arc of a Circle. From relation (1),
§ 56, it follows that
s = r8.
That is (Fig. 46), if a central angle is measured
in radians, and if its intercepted arc and the
radius of the circle are measured in terms of
the same unit, then
length of arc = radius x central angle in radians.
r~ EXERCISES
1. Express the following angles in radians :
25°, 145°, 225°, 300°, 270°, 450°, 1150°.
* 2. Express in degrees the following angles :
■K 7 IT blT 5TT
— , — , , u 7T, .
4' 6 6 '4
* 3. A circle has a radius of 20 inches. How many radians are there in
an angle at the center subtended by an arc of 25 inches ? How many
degrees are there in this same angle ? Ans.  r ; 71° 37' approx.
— i 4. Find the radius of a circle in which an arc 12 inches long subtends
an angle of 35°.
""" 5. The minute hand of a clock is 4 feet long. How far does its ex
tremity move in 22 minutes ?
6. In how many hours is a point on the equator carried by the rotation
of the earth on its axis through a distance equal to the diameter of the earth?
7. A train is traveling at the rate of 10 miles per hour on a curve of
half a mile radius. Through what angle has it turned in one minute ?
8. A wheel 10 inches in diameter is belted to a wheel 3 inches in
diameter. If the first wheel rotates at the rate of 5 revolutions per \\g
minute, at what rate is the second rotating? How fast must the former
rotate in order to produce 6000 revolutions per minute in the latter ?
58. Angular Measurement in Artillery Service. The
divided circles by means of which the guns of the United States Field
Artillery are aimed are graduated neither in degrees nor in radians, but
in units called mils. The mil is defined as an angle subtended by an arc
of ^^q of the circumference, and is therefore equal to
2tt 3.1416
6400 3200
0.00098175 =(0.001  0.00001825) radian.
VII, § 58] TRIGONOMETRIC RELATIONS
77
The mil is therefore approximately one thousandth of a radian.
(Hence its name.)*
Since (§57)
length of arc = radius x central angle in radians,
it follows that we have approximately
length of arc = x central angle in mils ;
1000
i.e. length of arc in yards a (radius in thousands of yards) • (angle
in mils). The error here is about 2 % .
Example 1. A battery occupies a front of 60 yd. If it is
at 5500 yd. range, what angle does it subtend (Fig. 47)? We
have, evidently,
angle = —= 11 mils.
5.5
Example 2. Indirect Fire, t A battery
posted with its right gun at G is to open fire on
a battery at a point T, distant 2000 yd. and in
visible from G (Fig. 48). The officer directing
tfie fire takes post at a point B from which both
the target T and a church spire P, distant
3000 yd. from <?, are visible. B is 100 yd. at
the right of the line 6? T and 120 yd. at the
right of the line GP and the officer finds by
measurement that the angle PBT contains
3145 mils. In order to train the gun on the
P target the gunner must set off the angle PG T
on the sight of the piece and then move the gun
Fig. 48
* To give an idea of the value in mils of certain angles the following has
been taken from the Drill Regulations for Field Artillery (1911), p. 164:
" Hold the hand vertically, palm outward, arm fully extended to the front.
Then the angle subtended by the
width of thumb is 40 mils
width of first finger at second joint is . ; . . .40 mils
width of second finger at second joint is .... 40 mils
width of third finger at second joint is 35 mils
width of little finger at second joint is 30 mils
width of first, second, and third fingers at second joint is . 115 mils
These are average values."
, t The limits of the text preclude giving more than a single illustration of
the problems arising in artillery practice. For other problems the student is
referred to the Drill Regulations for Field Artillery (1911) , pp. 57, 61, 150164 ;
and to Andrews, Fundamentals of Military Service, pp. 153159, from which
latter text the above example is taken.
78 PLANE TRIGONOMETRY [VII, § 58
until the spire P is visible through the sight. When this is effected, the
gun is aimed at T.
Let F and E be the feet of the perpendiculars from B to GT and GP
respectively, and let B T' and BP' be the parallels to G T and GP that
pass through B. Then, evidently, if the officer at B measures the angle
PBT, which would be used instead of angle PG T were the gun at B in
stead of at G, and determines the angles TBT' = FTB and PBP 1 = EPB,
he can find the angle PG T from the relation
PGT = PBT = PBT TBVPBP*.
Now tan FTB = — , tan EPB = — .
TF PE
small compared with G T and GP respectively, the radian measure of the
angle is approximately equal to the tangent of the angle. Why ? Hence
we have
FB)
FTB = tan FTB
GT
EPB = tan EPB = —
GP
approximately.
Therefore TBT' = FTB = — radians = 50 mils,
2000
PBP 1 = EPB = i^ radians = 40 mils.
3000
Hence PGT = PBT  TBT'  PBP 1
= 3145  50  40
= 3055 mils,
which is the angle to be set off on the sight of the gun.
Hence from the situation indicated in Fig. 48 we have the following
rule :
(1) Measure in mils the angle PBT from the aiming point P to the
target T as seen at B.
(2) Measure or estimate the offsets FB and EB in yards, the range
G T and the distance GP of the aiming point P in thousands of yards.
(3) Compute in mils the offset angles by means of the relations
TBT' = FTB,
PBP' = EPB,
TBT' = ^ B ~
GT
PBP' = — •
GP
(4) Then the angle of deflection PGT is equal to the angle PBT
diminished by the sum of the offset angles.
VII, § 59] TRIGONOMETRIC RELATIONS 79
EXERCISES
1. A battery occupies a front of 80 yd. It is at 5000 yd. range.
What angle does it subtend ?
2. In Fig. 48 suppose PBT = 3000 mils, FB = 200 yd., G T = 3000 yd.,
EB = 150 yd., GP = 4000 yd. Find the number of mils in PG T.
3. A battery at a point G is ordered to take a masked position and be
ready to fire on an indicated hostile battery at a point T whose range is
known to be 2100 yd. The battery commander finds an observing station
B, 200 yd. at the right and on the prolongation of the battery front, and
175 yd. at the right of PG. An aiming point P, 5900 yd. in the rear, is
found, and PBT is found to be 2600 mils. Find PG T.
4. A battery at a point G is to fire on an invisible object at a point T
whose range is known to be 2000 yd. A battery commander finds an
observing station B, 100 yd. at the right of G T and 150 yd. at the right
of GP. The aiming point P is 1500 yd. in front and to the left of G T.
The angle TBP contains 1200 mils. Find PG T.
59. The Sine Function. Let us trace in a general way the
variation of the function sin 6 as 6 increases from 0° to 360°.
For this purpose it will be convenient to think of the distance
r as constant, from which it follows that
the locus of P is a circle. When 6 = 0°, the
point P lies on the #axis and hence the
ordinate is 0, i.e. sin 0° = 0/r = 0. As 6
increases to 90°, the ordinate increases
until 90° is reached, when it becomes equal
to r. Therefore, sin 90° = r/r = 1. As FlG 49
increases from 90° to 180°, the ordinate de
creases until 180° is reached, when it becomes 0. Therefore
sin 180° = 0/r = As $ increases from 180° to 270°, the ordi
nate of P continually decreases algebraically and reaches its
smallest algebraic value when = 270°. In this position the
ordinate is — r and sin 270° = — r/r = — 1. When enters
the fourth quadrant, the ordinate of P increases (algebraically)
until the angle reaches 360°, when the ordinate becomes 0.
80
PLANE TRIGONOMETRY
[VII, § 59
Hence, sin 360° = 0. It then appears that :
as 6 increases from 0° to 90°, sin increases from to 1 ;
as increases from 90° to 180°, sin 6 decreases from 1 to ;
as increases from 180° to 270°, sin decreases from to — 1 ;
as 6 increases from 270° to 360°, sin 6 increases from — 1 to 0.
It is evident that the function sin 6 repeats its values in the
same order no matter how many times the point P moves
around the circle. We express this fact by saying that the
function sin 6 is periodic and has a period of 360°. In symbols
this is expressed by the equation
sin [8 + n • 360°] = sin 9,
where « is any positive or negative integer.
The variation of the function sin 6 is well shown by its
graph. To construct this graph proceed as follows : Take a
system of rectangular axes and construct a circle of unit radius
Fig. 50
with its center on the #axis (Fig. 50). Let angle XM 4 P = 0.
Then the values of sin 6 for certain values of 6 are shown in
the unit circle as the ordinates of the end of the radius drawn
at an angle 6.
e
30°
45°
60°
90°
sin
MiP t
M t P t
M Z P Z
M 4 P 4
...
Now let the number of degrees in be represented by dis
tances measured along OX. At a distance that represents 30°
erect a perpendicular equal in length to sin 30° ; at a distance
VII, § 60] TRIGONOMETRIC RELATIONS
81
that represents 60° erect one equal in length to sin 60°, etc.
Through the points 0, P l9 P 2 , — draw a smooth curve ; this
curve is the graph of the function sin 0.
If from any point P on this graph a perpendicular PQ is
drawn to the icaxis, then QP represents the sine of the angle
represented by the segment OQ.
Since the function is periodic, the complete graph extends
indefinitely in both directions from the origin (Fig. 51).
1&*X
ilar to those
60. The Cosine Function. By arguments s
used in the case of the sine function we may show that :
as 8 increases from 0° to 90°, the cos 6 decreases from 1 to ;
as increases from 90° to 180°, the cos decreases from to — 1 ;
as 6 increases from 180° to 270°, the cos increases from — 1 to ;
as 6 increases from 270° to 360°, the cos increases from to 1.
The graph of the function is readily constructed by a method
Fig. 52
similar to that used in the case of the sine function. This is
illustrated in Fig. 52.
The complete graph of the cosine function, like that of the
sine function, will extend indefinitely from the origin in both
82
PLANE TRIGONOMETRY
[VII, § 60
directions (Fig. 53). Moreover cos 6, like sin 6, is periodic and
has a period of 360°, i.e.
COS [6 4 71 • 360°] as cos 6,
where n is any positive or negative integer.
Y
61. The Tangent Function. In order to trace the varia
tion of the tangent function, consider a circle of unit radius
with^its center at the origin of a system of rectangular axes
(Fig. 54). Then construct the tangent to
this circle at the point M(l, 0) and let P
denote any point on this tangent line. If
angle MOP = 0, we have tan 6 = MP/OM
ae MP/1 = MP, i.e. the line MP represents
tan0.
Now when $ = 0°, MP is 0, i.e. tan 0° is 0.
As the angle 6 increases, tan 6 increases. As
approaches 90° as a limit, MP becomes
infinite, i.e. tan 6 becomes larger than any number whatever.
At 90° the tangent is undefined. It is sometimes convenient
to express this fact by writing
tan 90° =oo.
However we must remember that this is not a definition for
tan 90°, for oo is not a number. This is merely a short way of
saying that as approaches 90° tan becomes infinite and
that at 90° tan is undefined.
Thus far we have assumed to be an acute angle approach
ing 90° as a limit. Now let us start with as an obtuse angle
Fig. 54
VII, § 61] TRIGONOMETRIC RELATIONS
83
and let it decrease towards 90° as a limit. In Fig. 55 the line
MP' (which is here negative in direction) represents tan 0.
Arguing precisely as we did before, it is
seen that as the angle approaches 90°
as a limit, tan 6 again increases in magni
tude beyond all bounds, i.e. becomes infi
nite, remaining, however, always negative.
We then have the following results.
(1) When is acute and increases to
wards 90° as a limit, tan always remains
positive but becomes infinite. At 90° tan is undefined.
(2) When is obtuse and decreases towards 90° as a limit,
tan 6 always remains negative but becomes infinite. At 90°
tan 6 is undefined.
It is left as an exercise to finish tracing the variation of the
tangent function as 6 varies from 90° to 360°. Note that
tan 270°, like tan 90°, is undefined. In fact tan n • 90° is unde
fined, if n is any odd integer.
Fig.
Fig. 56
To construct the graph of the function tan 6 we proceed
along lines similar to those used in constructing the graph of
sin 6 and cos 0. The following table together with Fig. 56
illustrates the method.
84
PLANE TRIGONOMETRY
[VII, § 61
e
0°
30°
45°
60°
90°
120°
135°
150°
180°
210°
tan
MP X
MP 2
MP Z
undefined
MP A
MP b
MP 6
ilfP 7 =0
MP X
It is important to notice that tan 0, like sin 6 and cos 0, is
periodic, but its period is 180°. That is
tan(e + n180 o )=tan6,
where w is any positive or negative integer.
X
EXERCISES
1. What is meant by the period of a trigonometric function ?
2. What is the period of sin ? cos ? tan ?
3. Is sin defined for all angles ? cos ?
4. Explain why tan is undefined for certain angles. Name four
angles for which it is undefined. Are there any others ?
5. Is sin (0 + 360°) = sin ?
6. Is sin (0 + 180°) = sin ?
7. Is tan ( + 180°) = tan ?
8. Is tan (0 + 360°) = tan'0 ?
Draw the graphs of the following functions and explain how from the
graph you can tell the period of the function :
9. sin0. 11. tan0. 13. sec0.
10. cos0. 12. csc0. 14.' ctn0.
Verify the following statements :
15. sin90° + sin270° = 0. 18. cos 180° + sin 180° = 1.
16. cos 90° + sin0° = 0. 19. tan 360° + cos 360° = 1.
tan 1 80° + cos 1 80° =  1 . 20 . cos 90° + tan 180° I sin270^ = 1.^
21. Draw the graphs of the functions sin 0, cos 0, tan 0, making use of
a table of natural functions. See p. 112.
\2fc) Draw the curves y = 2 sin ; y = 2 cos ; y = 2 tan 6.
23. Draw the curve y = sin + cos 0.
24. From the graphs determine values of for which sin = \ ; sin
= 1 ; tan = 1; cos = \ ; cos = 1.
VII, § 63] TRIGONOMETRIC RELATIONS
85
62. The Trigonometric Functions of — 9. Draw the angles
6 and — 0, where OP is the terminal line of and OP is the
terminal line of — 6. Figure 57 shows an angle 6 in each of
r
Fig 57
the four quadrants. We shall choose OP = OP and («, y) as
the coordinates of P and (x', y') as the coordinates of P'. In
all four figures
t! =» x, y' =  y, r' = r.
Hence
sin(0) = ^ = :^ = sin0,
r r
cos ( — 6) m — ==  = cos 6,
r' r
_?/
y —
tan (  0) = 2 = —a =  tan (9.
Also,
esc ( — 6) = — esc 6 ; sec ( — 0) = sec ; ctn ( — 6) = — ctn 0.
The above results can be stated as follows : The functions of
— 6 equal numerically the like named functions of 6. The
algebraic sign, however, will be opposite except for the cosine
and secant.
Example, sin 10° = sin 10°, cos 10° = cos 10°, tan10°= tan 10°.
63. The Trigonometric Functions of 180° + 6. Similarly,
the following relations hold :
sin (180° + 0) = — sin 0, esc (180° + 6) =  esc 0,
cos (180° + 6) =  cos 0, sec (180° + 6) =  sec 6,
tan (180° + 6) = tan 0, ctn (180° + 6) = ctn 0.
The proof is left as an exercise.
86/ PLANE TRIGONOMETRY [VII, § 64
64. Summary. An inspection of the results of §§ 2728,
6263 shows :
1. Each f miction of — or 180° ± is equal in absolute value
(but not always in sign) to the same function of 0.
2. Each function of 90° — is equal in magnitude and in sign
to the corresponding cofunction of 6.
These principles enable us to find the value of any function
of any angle in terms of a function of a positive acute angle
(not greater than 45° if desired) as the following examples
show.
Example 1. Reduce cos 200° to a function of an angle less than 45°.
Since 200° is in the third quadrant, cos 200° is negative. Hence
cos 200° =  cos 20°. Why ?
Example 2. Reduce tan 260° to a function of an angle less than 45°.
Since 260° is in the third quadrant, tan 260° is positive. Hence
tan 260° = tan 80° = ctn 10° (§ 27).
Example 3. Reduce sin (— 210°) to a function of a positive angle
less than 45°.
From § 62 we know sin — 210° = — sin 210°.
Considering the positive angle 210°, we have
sin  210° =  sin 210° =  [  sin 30°] = sin 30°.
EXERCISES
Reduce to a function of an angle not greater than 45° :
1. sin 163°. 5. esc 901°.
2. cos (110°). *"+ i. ctn (1215°). + 
Ans. sin 20°. 7> tan 840°.
> 3. sec (265°). 8. sin 510°.
4. tan 428°. tX— tv.
Eind without the use of tables the values of the following functions :
— >9. cos 570°. 11. tan 390°. 13. cos 150°.
10.' sin 330°. ^* 12. sin 420°. 14. tan 300°.
Reduce the following to functions of positive acute angles :
^15. sin 250°. T* 18. sec (245°).
Ans. — sin 70° or — cos 20°. 19. C sc(— 321°).
16. cos 158°. 20. sin 269°.
17. tan (389°).
VII, § 64] TRIGONOMETRIC RELATIONS 87
Prove the following relations from a figure :
(a) sin (90° + 0) = cos 0.
(O
sin (180° + 0) = — sini
cos (90° + 0) = — sin 0.
cos (180°+ 0) = cos
tan (90° + 0) = — ctn 0.
tan (180° + 0)= tan0.
esc (90° + 0)=sec0.
csc(18O° + 0) = — csci
sec (90° + 0) =  esc 0.
sec (180° + 0) = seci
ctn (90° + 0) =  tan 0.
ctn (180° + 0)=ctn0.
(b) sin (180° 6)=sm0.
(<*)
sin (270° 0) =cos
cos (180° — 6) = — cos 6.
cos (270° —0) =  sin i
tan (180°  0) = tan0.
tan (270°  0) = ctn 0.
esc (180° — d) = esc 6.
esc (270° — 0) = — sec
sec (180° — d) = — sec0.
sec (270°  0) =  esc
ctn (180° 0) =  ctn 6.
ctn (270°  0) = tan 0.
(e) sin (270° + 0) =  cos 0.
cos (270° + 0) = sin 0.
tan (270° + 0) =  ctn 0.
esc (270° + 0) =  sec 0.
sec (270° + 0) = esc 0.
ctn (270° + 0) =  tan 0.
t J 4t^r
Hn ik
a o
B
e ta csa
—
fcuk
MlljJIlMllH
CHAPTER VIII
TRIGONOMETRIC RELATIONS (Continued)
^5. Trigonometric Equations. An identity, as we have
seen (§ 26), is an equality between two expressions which is
satisfied for all values of the variables for which both expres
sions are defined. If the equality is not satisfied for all
values of the variables for which each side is defined, it is
called a conditional equality, or simply an equation. Thus
1 — cos = is true only if = n • 360°, where n is an integer.
To solve a trigonometric equation, i.e. to find the values of
for which the equality is true, we usually proceed as follows.
1. Express all the trigonometric functions involved in terms
of one trigonometric function of the same angle.
2. Find the value (or values) of this function by ordinary
algebraic methods.
3. Eind the angles between 0° and 360° which correspond to
the values found. These angles are called particular solutions.
4. Give the general solution by adding n • 360°, where n is
any integer, to the particular solutions.
Example 1. Find 6 when sin 6 = $.
The particular solutions are 30° and 150°.
30° + n ■ 360°, 150° + n • 360°.
The general solutions are
Example 2. Solve the equation tan 6 sin d — sin = 0.
Factoring the expression, we have sin (tan 6 — Y)= 0. Hence we
have sin = 0, or tan 6 — 1 = 0. Why ?
The particular solutions are therefore 0°, 180°, 45°, 225°. The genera!
solutions are n . 360°, 180° + n . 360°, 45° + n • 360°, 225° + n • 360°.
88
2.
sin = — — •
2 ,
3.
2
4.
2
5.
tan0 = — 1.
JL
ctn 0=1.
16.
2 sin = tan 0.
VIII, § 66] TRIGONOMETRIC RELATIONS 89
EXERCISES
Give the particular and the general solutions of the following
equations :
tJq 7. sec — 2.
2 8. tan = 0.
Vi 9. sec 2 = 2.
10. sin 2 = .
11. cos0= — £.
12. csc 2 = f
/l3. 4 sin — 3 esc = 0.
1 14. 2 sin cos 2 = sin 0.
)l5. cos f sec = f .
^Irw. Particular solutions : 0°, 180°, 60°, 300°.
17. 3 sin + 2 cos = 2. /l8. 2 cos 2 0—1 = 1 — sin 2 0.
Inverse Trigonometric Functions. The equation
x — sin y (1)
may be read :
y is an angle whose sine is equal to x,
a statement which is usually written in the contracted form
y = arc sin x.* (2)
For example, x = sin 30° means that x = \, while y = arc sin i
means that y = 30°, 150°, or in general (n being an integer),
30° + n • 360° ; 150° + n • 360°.
Since the sine is never greater than 1 and never less than
— 1, it follows that —l_\x—\l. It is evident that there is
an unlimited number of values ofy = arc sin x for a given value
of x in this interval.
We shall now define the principal value Arc sin x f of arc sin x,
distinguished from arc sin x by the use of the capital A, to be
* Sometimes written y = sin 1 x. Here — 1 is not an algebraic exponent,
but merely a part of a functional symbol. When we wish to raise sin x to
the power — 1, we write (sin x)}.
t Sometimes written Sini x, distinguished from sin 1 x by the use of the
capital S.
90
PLANE TRIGONOMETRY
[VIII, § 66
the numerically smallest angle whose sine is equal to x. This func
tion like arc sin x is denned only for those values of x for
which
The difference between arc sin x and Arc sin x is well illus
trated by means of their graph. It is
evident that the graph ofy = arc sin x,
i.e. x = sin y is simply the sine curve
with the role of the x and y axes inter
changed. (See Fig. 58.) Then for every
admissible value of x, there is an un
limited number of values of y ; namely,
the ordinates of all the points P 1} P 2 , •, in
which a line at a distance x and parallel
to the 2/axis intersects the curve. The
singlevalued function Arc sin x is repre
sented by the part of the graph between
M and N,
Similarly arc cos x, defined as " an angle whose cosine is x,"
has an unlimited' number of values for
every admissible value of x(— 1 f^ x < 1)
We shall define the principal value Arc
cos x as the smallest positive angle whose
cosine is x. That is.
Fig. 58
^ Arc cos x <^ 7r.
Figure 59 represents the graph of y = arc
cos x, and the portion of this graph between
M and N represents Arc cos x.
Similarly we write x = tan y as y = arc
tan x, and in the same way we define the
symbols arc ctn x ; arc sec x ; arc esc x.
The principal values of all the inverse trigonometric functions
are given in the following table.
Y
2tt
37T
P»
N \
IT
7T
2
ft
M
1
Pi
1 X
V= arc cos x
y=Arc cos x
Fig. 59
VIII, § 66] TRIGONOMETRIC RELATIONS
91
y
Arc sin x
Arc cos x
Arc tan x
Range of x
lgx^ 1
l^a^l
all real values
Range of y
7T . 7T
to —
2 2
tO 7T
to —
2 2
x positive
1st Quad.
1st Quad.
1st Quad.
x negative
4th Quad.
2d Quad.
4th Quad.
Arc ctn x
Arc sec x
Arc cscx
Range of x
all values
x^l orx^ 1
a;^lorx2l
Range of y
OtOT
tO 7T
to —
2 2
x positive
1st Quad.
1st Quad.
1st Quad.
x negative
2d Quad.
2d Quad.
4th Quad.
In so far as is possible we select the principal value of each
inverse function, and its range, so that the function is single
valued, continuous, and takes on all possible values. This ob
viously cannot be done for the Arc sec x and for Arc esc y.
EXERCISES
1. Explain the difference between arc sin x and Arc sin x.
>^. Find the values of the following expressions :
\(a) Arc sin \. (d) Arc tan — 1.
(e) arc cos
V3
(/) Arc cos 22.
>>^&) arc sin \.
(c) arc tan 1. 2
S^What is meant by the angle it ? tt/4 ?
4. Through how many radians does the minute hand of a watch turn
in 30, minutes ? in one hour ? in one and one half hours ?
6. For what values of x are the following functions defined :
y\d) arc sin x ? ^/($) arc tan x ? _^ e ) arc sec x ?
(6) arc cos x ? (d) arc ctn x ? (/) arc esc x ?
6. What is the range of values of the functions :
(or) Arc. sin x ? (c) Arc tan x ? (e) Arc sec x.
(6) Arc cos x ? (d) Arc ctn x ? (/) Arc esc x ?
fa I
<0
TW
92
PLANE TRIGONOMETRY
[VIII, § 66
7. Draw the graph of the functions :
(a) arc sin x. (c) arc tan x. (e) arc sec x.
(&) arc cos x. (d) arc etna;. (/) arc esc x.
8. Find the value of cos (Arc tan f).
Hint. Let Arc tan f = 6. Then tan d = £ and we wish to find the
value of cos 6.
: 9. Find the values of cos (arc tan f ) . !^V> •
TIC Find the value of the following expressions :
(a) sin (arc cos ). *"* (c) cos (Arc cos T 5 ^). (e) sin (Arc sin \).
(6) sin (arc sec 3). {$) sec (Arc esc 2). (/) tan (Arc tan 5) .
11. Prove that Arc sin (2/5)= Arc tan (2/V21)
12. Find x when Arc cos (2 x 2  2 x) = 2 tP/3. \ ^
Find the values of the following expressions :
13. cos [90°— Arc tan f].
j£f 1^1
f
14. sec [90° — Arc sec 2].
15. tan [90°  Arc sin T \].
67. Projection. Consider two directed lines p and q in a
plane, i.e. two lines on each of which, one of the directions
has been specified as positive (Fig. 60). Let A and B be
any two points on p and let A', B' be the points in which per
Fig. 00
pendiculars to q through A and B, respectively, meet q. The
directed segment A'B' is called the projection of the directed seg
ment AB on q and is denoted by
A'B' = proj ff AB.
In both figures AB is positive. In the first figure A'B' is posi
tive, while in the second figure it is negative.
As special cases of this definition we note the following :
VIII, § 67] TRIGONOMETRIC RELATIONS 93
1. If p and q are parallel and are directed in the same way,
we have
proj, AB = AB.
2. If p and q are parallel and are directed oppositely, we
have
proj ff AB = — AB.
3. If p is perpendicular to q, we have
proj, AB = 0.
It should be noted carefully that these propositions arc true
no matter how A and B are situated on p.
We may now prove the following important proposition :
If A and B are any two points on a directed line p, and q is
any directed line in the same plane with p, then we have both
in magnitude and sign
(1) projg AB = AB ■ cos (pq)* = AB . cos (qp).
We note first from § 8 that (pq)+ (qp) = + n 360°, where
n is any integer. Hence from § 64, cos (jxj) = cos (qp). Two
cases arise.
Jt
T22
Fig. 61
Case 1. Suppose AB is positive, i.e. it has the same direc
tion as p.
Through A draw a line q^ parallel to q and with the same
direction. [It is evident that we may assume without loss of
generality that q is horizontal and is directed to the right.]
Let A'B' be the projection of AB on q and let BB' meet q x
in B x . Then by the definition of the cosine we have
AB
——± = cos (qip) = cos (pqi) = cos (qp) = cos ( pq)
AB
* (pq) represents an angle through which p may be rotated in order to
make its direction coincide with the direction of q ; similarly for (qp).
94
PLANE TRIGONOMETRY
[VIII, § 67
in magnitude and sign. Hence
AB± = AB ■ cos (pq) = AB • cos (qp).
But AB X = A'B' = proj 3 AB.
Therefore proj tf AB = AB • cos (pp) = AB • cos (qp).
Case 2. Suppose AB is negative.
If AB is negative, BA is positive and we have from Case 1,
B'A! = BA • cos (pq) = BA • cos (qp).
Changing the signs of both members of this equation, we have
A'B' = AB • cos (})q)= AB • cos (qp).
The special cases 1, 2, 3, are obtained from formula (1)
by placing (qp) or (pq) equal to 0°, 180°, 90° respectively.
Theorem. If A, B, C are any three points in a plane, and I
is any directed line in the plane, the algebraic sum of the projec
tions of the segments AB and BC on I is equal to the projection
of the segment AC on I.
As a point traces out the path from A to B, and then from
B to C (Fig. 62), the projection of the point traces out the
segments from A' to B' and then from B'
to C. The tjjjft result of this motion is a
motion from A' to O which represents
the projection of AC, i.e.
A'B' + B'C = A' C.
I
/
— ' — ■
z^
B
ti
C
>^
A'
(
i'
h
'
EXERCISES
1. What is the projection of a line segment upon a line I, if the line
segment is perpendicular to the line I ?
2. Find proj x ^4JB and proj^l?* in each of the following cases, if a
denotes the angle from the xaxis to AB.
(a) AB = 5, a = 60°. (c) AB = 6, a = 90°.
(6)^45 = 10, a = 300°. (d) AB = 20, a = 210°.
* Proj x AB and proj,, AB mean the projections of AB on the xaxis and
the yaxis, respectively.
<^ w
<^
VIII, § 68] TRIGONOMETRIC RELATIONS 95
3. Prove by means of projection that in a triangle ABC
a—b cos C f c cos B.
4. If projj. AB = 3 and proj„ AB = —4, find the length of AB.
5. A steamer is going northeast 20 miles per hour. Hots fast is it
going north ? going east ?
6. A 20 lb. block is sliding down a 15° incline. Find what force
acting directly up the plane will just hold the block, allowing ope half a
pound for friction.
7. Prove that if the sides of a polygon are projected in order upon any
given line, the sum of these projections is zero.
Fig. 63
The Addition Formulas. We may now derive formulas
for sin (a f /3), cos (a f ft), and tan (a + ft) in terms of func
tions of a and ft. To this end
let P(x, y) be any point on the
terminal side of the angle a (the
initial side being along the posi
tive end of the a>axis and the
vertex being at the origin). The
angle a + ft is then obtained by
rotating OP through an an^le
ft. If P' (x', y') is the new Sta
tion P after this rotation and
OP = OP' = r, we have sin (a f ft) = £ , cos (a + ft) =  , by
v r
definition. Our first problem is, therefore, to find x' and y' in
terms of x, y, and ft.
In the figure OMP is the new position of the triangle OMP
after rotating it about through the angle ft. Now,
x' = proj x OP' ss proj x OM' + proj x M'F
= xcosft + ycos(ft + ^\
= x cos ft — y sin ft.
96 PLANE TRIGONOMETRY [VIII, § 68
Similarly,
y> = proj, OP' = proj, OM' + proj tf M'F
= x cos(? ft\+ y cos ft
= x sin ft + y cos /?.
Hence, , , ~ x y' x ,+ n . V n
' sm (« + j3)=V= sr$/3+^ cos £
r r
= sin a cojr [
or (1) m sin (a + P) = sin a co# p f cos a sin p.
Also
cos
s(« + £) = ^ = ^osft^ sin/?.
or (2) cos (a + p) = cos a cos p — sin a sin p.
Further we have
tan (a 4 B) = S * n ( a ~*~ ® = sni g cos ft + cos <* sin ft
cos (a 4 ft) cos a cos ft — sin a sin /J
Dividing numerator and denominator by cos a cos ft, we have
(3) • tan(a+B)= tan * + tan P.
w v K; 1  tan a tan p
Furthermore, by replacing ft by — ft in (1), (2), and (3), and
recalling that
sin (— ft) = — sin ft, cos (— ft) = cos ft, tan (— ft) = — tan ft,
we obtain ^^fc_
(4) sin (a — P) = sin a cmf$ — cos a sin p,
(5) cos (a — P) = cos a cm$ ■+ sin a sin p,
(6) tan (a.  tt = tan o^ tan p
tan (a  p) = — y — r
v r/ 1 + t*n a tan p
EXERCISES
Expand the iollowing :
*±r sin (45° + «) = 3. cos (60° + a) = 5. sin (30°  45°) =
—=«. tan (30°  0) = 4. tan (45° + 60°) = . 6. cos (180°  45°) =
7. What do the following formulas become if « = /3 ?
sin (« + (S)= sin a cos /3 + cos a sin p. t (a A 8 s )— tan a + tan P .
sin (a — /3) = sin a cos /3 — cos a sin 0. * 1 — tan a tan /3
cos (a + /3) = cos a cos — sin a sin /S. . , _ q\ _ tan a — tan g (
cos (a — /3) = cos a cos p + sin a sin j8. 1 + tan a tan /3
VIII, § 68] TRIGONOMETRIC RELATIONS 97
8. Complete the following formulas :
sin 2 a cos a + cos 2 a sin a — tan 2 a + tan a _
sin 3 a cos a — cos 3 a sin a = 1 — tan 2 a tan a
^* Prove sin 75° = V ^ + 1 , cos 75° = V ^ ~ 1 , tan75° = V g + 1 
2V2 2V2 V31
10. Given tan a = f , sin ft = T 5 ^, and a and ft both positive acute angles,
find the value of tan (a + ft); shr(a~— ft); cos (a + ft); tan (a — ft).
,. »r1 1. Prove that
(a) cos (60° + a) + sin (30° + a) = cos a.
* ( 6) sin (60° + 0)  sin (60°  6)  sin 0.
(c) cos (30° + 0) cos (30°  0)=  sin 6.
(d) cos (45° + 6) + cos (45°  0) = V2 • cos 0.
> (e) sin 1 a +  ) + sin ( a — — j = sin a.
(/) cos ( a +  ) + cos (a — ) = V3 • cos a.
~* — 12. By using the functions of 60° and 30° find the value of sin 90° ;
cos 90°.
13. Find in radical form the value of sin 15° ; cos 15° ; tan 15° ;
sin 105° ; cos 105° ; tan 105°.
14. If tan a = , sin ft = T 5 T , and a is in the third quadrant while ft is
in the second, find sin (a ± ft) ; cos (a ± ft) ; tan (a ± ft).
Prove the following identities : <^"~"
15 sin (a + ft) _ tan a + tan ft _ 16 sin 2 a , cos 2 a _ sm 3 a
sin (a — ft) tana — tan ft sec a esc a
17 tana tan (a ft) = tan ^ 19. ( a ) sin ( 180 o _ 9)  s i n $m
1 + tan a tan (a— ft) (6) cos (180°  6) =  cos 0.
x 18. tan(0±45°) + ctn(0T45°)=O. (c) tan (180°  6) =  tan 0.
20. cos (a f ft) cos (a — ft) = cos 2 a — sin 2 ft.
21. sin (a + ft) sin (a — ft) = sin 2 a  sin 2 ft.
22. ctn(« + /9) = ctnttctn g 1 . 23. ctn (a  ft) = Ctn " ctn ** + * .
ctna + ctnft ctnft — etna
24. Prove Arc tan £ + Arc tan  = ?r/4.
[Hint : Let Arc tan \ = x and Arc tan \ = y. Then we wish to prove
x + y = ir/4, which is true since tan (x + y)= 1.]
25. Prove Arc sin a + Arc cos a =  if < a < 1.
p
26. Prove Arc sin T * 7 f Arc sin  = Arc sin .
i H
A
98 x PLANE TRIGONOMETRY [VIII, § 68
27. Prove Arc tan 2 + Arc tan £ as ir/2.
28. Prove Arc cos § + Arc cos (— T 5 y) = Arc cos (— f).
29. Prove Arc tan T 8 5 + Arc tan f = Arc tan f £.
30. Find the value of sin [Arc sin  + Arc ctnf ].
 31. Find the value of sin [Arc sin a + Arc sin 6] if < a < 1, < b < 1.
32. Expand sin (x + y + z) ; cos(x + y + z).
[Hint : x + y + z =(x + y)+ z.]
33. The area i of a triangle was computed from the formula
A = I ab sin 0. If an error c was made in measuring the angle 0, show that
the corrected area A' is given by the relation.^.' = A(cos e + sin e ctn 6).
69. Functions of Double Angles. In this and the follow
ing articles (§§ 6971) we shall derive from the addition
formulas a variety of other relations which are serviceable in
transforming trigonometric expressions. Since the formulas
for sin (a + fi) and cos (a + /?) are true for all angles a and (3,
they will be true when /? = a. Putting /3 = a, we obtain
(1) sin 2 a = 2 sin a cos a,
(2) cos 2 a = cos 2 a — sin 2 a.
Since sin 2 a + cos 2 a — 1, we have also
(3) cos 2 a = 1  2 sin 2 a
(4) =2cos 2 al.
Similarly the formula for tan (a + ft) (which is true for all
angles a, ft, and a+ft which have tangents) becomes, when ft=a,
(5) tan2q= 2tana ,
v ; ltan 2 a
which holds for every angle for which both members are denned.
The above formulas should be learned in words. For ex
ample, formula (1) states that the sine of any angle equals
twice the sine of half the angle times the cosine of half the
angle. Thus sin6^ = 2 sin3« cos3^,
2 tan 2 x
tan 4 x —
ltan 2 2x'
cos x = cos 2  — sin 2 >
VIII, § 70] TRIGONOMETRIC RELATIONS 99
70. Functions of Half Angles. From (3), § 69, we have
Therefore
2sin 2 £=l — cos a.
(6)
«in«_  /I cos a
° m 2 _± \ 2
From (4), § 69, we have
2 cos 2  = 1 + cos a.
Therefore
(7) cos« = ± /±f5^.
Formulas (6) and (7). are at once seen to !
«. Now, if we divide formula (6) by formula (7), we obtain
/QX * a /l — cos a
(8) tan  = ± \/ ,
v } 2 ' V'l + cosa'
which is true for all angles a except n • 180°, where n is any
odd integer.
Example. Given sin^. =— 3/5, cos 4 negative ; find sin (A/2).
Since the angle A is in the third quadrant, A/2 is in the second or
fourth quadrant, and hence sin (A/2) may be either positive or negative.
Therefore, since cos A = — 4/5, we have
2 \ 2 «/Tn 10
VTo io
EXERCISES
Complete the following formulas and state whether they are true for
all angles :
1. sin 2 a = 3  tan 2 a — 5. cos " =
A
2. cos2a= (three forms). 4 s in= 6. tan  =
2 2
7. In what quadrant is 0/2 if 6 is positive, less than 360°, and in the
second quadrant ? third quadrant ? fourth quadrant ?
8. Express cos 2 a in terms of cos 4 a.
9. Express sin 6 x in terms of functions of 3 x.
100 PLANE TRIGONOMETRY [VIII, § 70
10. Express tan 4 a in terms of tan 2 a.
11. Express tan 4 a in terms of cos 8 a.
12. Express sin x in terms of functions of x/2.
13. Explain why the formulas for sin x and cos x in terms of functions
of 2 x have a double sign.
14. From the functions of 30° find those of 60°.
15. From the functions of 60° find those of 30°.
16. From the functions of 30° find those of 15°.
17. From the functions of 15° find those of 7°. 5.
18. Find the functions of 2 a if sin a = $ and a is in the second
quadrant.
19. Find the functions of a/2 if cos a =— 0.6 and a is in the third
quadrant, positive, and less than 360°.
20. Express sin 3 a in terms of sin a. [Hint : 3a = 2a + a.]
21. From the value of cos 45° find the functions of 22°. 5.
22. Given sin a = — and a in the second quadrant. Find the values of
(a) sin 2 a. (c) cos 2 a. (e) tan 2 a.
(6) sin". (d) cos?. (/) tan.
23. If tan 2 a =  find sin a, cos a, tan a if a is an angle in the third
quadrant.
Prove the following identities :
24. 1 + C0 *«=cto& 27. lcos2fl + sin2fl =tan
. sin a 2 . . 1 + cos 2 + sin 2
25.
26.
Tsin — cos] =1 — sin0. 28. sin + cos — = ± Vl + sina.
L.2 2J 22
cos2 + cos0 4l „ ctn ,, j 29 Be0 a + tan«=ten^ + ^V
sin20 + sin0 \4 2/
30. 2 Arc cos a; = Arc cos (2 x 2 — 1).
31. 2 Arc cosx = Arc sin (2 xVl — x 2 ).
32. tan [2 Arc tanx] = ^^. 34. tan [2 Arc sec x] = ± 2 '
1  x 2 J 2  x 2
33. cos [2 Arc tan x] = — x • /35^ *os (2 Arc sin a) = 1 — 2 a 2 .
1 +x 2
Solve the following equations
36. cos 2 x + 5 sin x = 3. 40. sin 2 2 x — sin 2 x ss f .
37. cos2x — sinx = \. 41. sin2x = 2cosx.
38. sin 2 x cos x = sin x. 42. 2 sin 2 2 x = 1 — cos2x.
39. 2sin 2 x + sin 2 2x = 2. 43. ctnx — csc2x — 1.
fa
VIII, § 71] TRIGONOMETRIC RELATIONS 101
44. A flagpole 50 ft. high stands on a tower 49 ft. high. At what dis
tance from the foot of the tower will the flagpole and the tower subtend
equal angles ?
45. The dial of a town clock h^s a diameter (jf JO ft. and its center is
100 ft. above the ground. At 1 what' distance from the foot of the tower
will the dial be
must be as large
most plainly viqbie f] £r he ajn^'fubWrded by the dial
as possible.]* ° ' •••• '
71. Product Formulas. From § 68 we have
sin (a\ (3) = sin a cos /? f cos a sin /3,
sin (a — /?) = sin a cos /? — cos a sin /?.
Adding, we get
(1) sin (a + p) + sin (a — /?) = 2 sin a cos /3.
Subtracting, we have
(2) sin (a + ft) — sin (« — p) = 2 cos a sin 0.
Now, if we let a f /? = P and a — ft = Q,
thell « = ^, = Z^$.
Therefore formulas (1) and (2) become
P + O P
sin P + sin Q = 2 sin ^ v cos —
2 2
Pi Q p
sin P — sin Q = 2 cos — — * sin —
2
Similarly, starting with cos (« + /?) and cos (a — /?) and per
forming the same operations, the following formulas result :
P 4 O P — O
cos P + cos Q = 2 cos — —* cos — —i,
A A
cos P — cos Q = — 2 sin J~ v sin — —^.
2 2
y^. In words :
the sum of two sines =
twice sin (half sum) times cos (half difference),
the difference of two sines =
twice cos (half sum) times sin (half difference),*
* The difference is taken, first angle minus the second.
102 PLANE TRIGONOMETRY [VIII, § 71
the sum of two cosines =
twice cos (half sum) times cos (half difference),
the difference of two cosines, =
minus twice sin (half sum) times sin (half difference). *
Example 1. Prove that ' a f
coB8a ; + co 8:a ?=ctn j a ,
sin 3 x + sin x
for all angles for which both members are defined.
cos 3 x + cos x _ 2 cos ^(3 x 4 x) cos (3 x — x) _ cos 2 x _ . 9
sin 3 x + sin x 2 sin £(3 x + x) cos \ (3 x — x) ~" sin 2 x ~~
Example 2. Reduce sin 4 x 4 cos 2 x to the form of a product.
We may write this as sin 4 x 4 sin (90° — 2x), which is equal to
2 sin Ix + WZ* cos txW + az 2 sin (45 „ + x) cos (3 x _ 45 „ } _
EXERCISES
Reduce to a product :
1. sin 4 — sin 2 0. 4. cos 2 + sin 2 0. 7. cos 3 x + sin 5 x.
2. cos + cos 3 0. 5. cos 3 — cos 6 0. 8. sin 20° — sin 60°.
3. cos 6 + cos 2 0. 6. sin (x f Ax) — sin x.
Show that
9. sin 20° + sin 40° = cos 10°. 12 s in 15° 4 sin 75° _ _ 6QO
10. cos 50° 4 cos 70° = cos 10°. ' sin 15°  sin 75° ~
11. sin75 ° sinl5 ° = tan 30°. 13. sin3 0sin5 = _ ^ 4 ,
cos 75° 4 cos 15° cos 3 — cos 5
Prove the following identities : , ^ "
"^ sm~$TT4 r 'Slfi~3 ft _ g^fl" 15 sin a + sin ft _ tan \ (a + ft)
cos 3 a — cos 4 a 2 ' sin a — sin ft tan £ (a — ft)
. .. cos a 4 2 cos 3 a 4 cos 5 a cos 3 a
id. = •
cos 3 a. + 2 cos 5 a 4 cos 7 a cos 5 a
_ cos a— cos ft _ _ tan ^(#4 ft) lg sin (n — 2) 4 sin nd _ .
cos «4 cos ft ctn£(a — ft) ' cos (n — 2) — cos nd
Solve the following equations :
■ 19. cos 4 cos 50 = cos 30. 22. sin 4 — sin 2 = cos 3 0.
20. sin 4 sin 5 = sin 3 0. 23. cos 7 — cos = — sin 4
21. sin 3 6 + sin 7 = sin 5 0.
*The difference is taken, first angle minus the second.
I . OW^&
VIII. § 71] TRIGONOMETRIC RELATIONS 103
MISCELLANEOUS EXERCISES
1. Reduce to radians 65°,  135°,  300°, 20°.
2. Reduce to degrees 7r, 3 ?r, — 2 w, 4 v radians.
3. Find sin (a — /3) and cos (a + /S) when it is given that a and /3 are
positiye and acute and tan a = f and sec /3 = *£.
4. Find tan (a + /S) and tan (a — /3) when it is given that tan a = \
and tan /S = .
5. Prove that sin 4 a = 4 sin a cos a — 8 sin 3 a cos a.
6. Given sin = —  y and in the second quadrant. Find sin 2
V5
cos 2 0, tan 2 0.
Prove the following identities :
. 7 . sin2«= 2tan " ■ 9. sec2« csc2a
1 + tan 2 a esc 2 a — 2
8. cos2 ( , = 1  tan2 ^. 10. tan«= sin2a
1 + tan 2 a 1 + cos 2 a
s~ 11. sin (a + /S) cos /3 — cos (a + /3) sin = sin a.
^£ll. sin 2 a + sin 2 j3 + sin 2 7 = 4 sin a sin sin 7, if a + /S + 7 = 180°.
1 + tan  c* , q.
cos a 2 _d J <~ ^
' lsta«"l_ta„!' "= r T\^P &
— \_ ten x. 4 ^_ 1
. A *• % <u*J~^&. ^ _ 1
4^»\A 5i_ .2
• *^c4^rv^ U^
6p
Jf^iilO *q
sp
OQ
0Ti + #f T#
^~ (ot +ft) ■* %*~oC(U>($ *6*>©^
u.
«
£_ ff ^^tu#^
m
T<. u v
ir.o
m
Ho'
5 R
M*
f * '
^ r
% *

To*
3
T?
fi^L^ I
/^/^C
TABLES
FOUR DECIMAL PLACES
106
[Moving the decimal poin
Squares of Numbers
t one place in N requires a corresponding move of two
places in N 2 ]
u
N 2
1
2
3
4
5
6
7
8
9
0.0
.0000
.0001
.0004
.0009
.0016
.0025
.0036
.0049
.0064
.0081
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
.0100
.0400
.0900
.1600
.2500
.3600
.4900
.6400
.8100
.0121
.0441
.0961
.1681
.2601
.3721
.5041
.6561
.8281
.0144
.0484
.1024
.1764
.2704
.3844
.5184
.6724
.8464
.0169
.0529
.1089
.1849
.2809
.3969
.5329
.6889
.8649
.0196
.0576
.1156
.1936
.2916
.4096
.5476
.7056
.8836
.0225
.0625
.1225
.2025
.3025
.4225
.5625
.7225
.9025
.0256
.0676
.1296
.2116
.3136
.4356
.5776
.7396
.9216
.0289
.0729
.1369
.2209
.3249
.4489
.5929
.7569
.9409
.0324
.0784
.1444
.2304
.3364
.4624
.6084
.7744
.9604
.0361
.0841
.1521
.2401
.3481
.4761
.6241
.7921
.9801
1.0
1.000
1.020
1.040
1.061
1.082
1.103
1.124
1.145
1.166
1.188
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
1.210
1.440
1.690
1.960
2.250
2.560
2.890
3.240
3.610
1.232
1.464
1.716
1.988
2.280
2.592
2.924
3.276
3.648
1.254
1.488
1.742
2.016
2.310
2.624
2.958
3.312
3.686
1.277
1.513
1.769
2.045
2.341
2.657
2.993
3.349
3.725
1.300
1.538
1.796
2.074
2.372
2.690
3.028
3.386
3.764
1.323
1.563
1.823
2.103
2.403
2.723
3.063
3.423
3.803
1.346
1.588
1.850
2.132
2.434
2.756
3.098
3.460
3.842
1.369
1.613
1.877
2.161
2.465
2.789
3.133
3.497
3.881
1.392
1.638
1.904
2.190
2.496
2.822
3.168
3.534
3.920
1.416
1.664
1.932
2.220
2.528
2.856
3.204
3.572
3.960
2.0
4.000
4.040
4.080
4.121
4.162
4.203
4.244
4.285
4.326
4.368
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.0
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
4.410
4.840
5.290
5.760
6.250
6.760
7.290
7.840
8.410
4.452
4.884
5.336
5.808
6.300
6.812
7.344
7.896
8.468
4.494
4.928
5.382
5.856
6.350
6.864
7.398
7.952
8.526
4.537
4.973
5.429
5.905
6.401
6.917
7.453
8.009
8.585
4.580
5.018
5.476
5.954
6.452
6.970
7.508
8.066
8.644
4.623
5.063
5.523
6.003
6.503
7.023
7.563
8.123
8.703
4.666
5.108
5.570
6.052
6.554
7.076
7.618
8.180
8.762
4.709
5.153
5.617
6.101
6.605
7.129
7.573
8.237
8.821
4.652
5.198
5.664
6.150
6.656
7.182
7.728
8.294
8.880
4.796
5.244
5.712
6.200
6.708
7.236
7.784
8.352
8.940
9.000
9.060
9.120
9.181
9.242
9.303
9.364
9.425
9.486
9.548
9.610
10.24
10.89
11.56
12.25
12.96
13.69
14.44
15.21
9.672
10.30
10.96
11.63
12.32
13.03
13.76
14.52
15.29
9.734
10.39
11.02
11.70
12.39
13.10
13.84
14.59
15.37
9.797
10.43
11.09
11.76
12.46
13.18
13.91
14.70
15.44
9.860
10.50
11.16
11.83
12.53
13.25
13.99
14.75
15.52
9.923
10.56
11.22
11.90
12.60
13.32
14.06
14.82
15.60
9.986
10.63
11.29
11.97
12.67
13.40
14.14
14.90
15.68
10.05
10.69
11.36
12.04
12.74
13.47
14.21
14.98
15.76
10.11
10.76
11.42
12.11
12.82
13.54
14.29
15.05
15.84
10.18
10.82
11.49
12.18
12.89
13.62
14.26
15.13
15.92
4.0
16.00
16.08
16.16
16.24
16.32
16.40
16.48
16.56
16.65
16.73
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
16.81
17.64
18.49
19.36
20.25
21.16
22.09
23.04
24.01
16.89
17.72
18.58
19.45
20.34
21.25
22.18
23.14
24.11
16.97
17.81
18.66
19.54
20.43
21.34
22.28
23.23
24.21
17.06
17.89
18.65
19.62
20.52
21.44
22.37
23.33
24.30
17.14
17.98
18.84
19.71
20.61
21.53
22.47
23.43
24.40
17.22
18.06
18.92
19.80
20.70
21.62
22.56
23.52
24.50
17.31
18.15
19.01
19.89
20.79
21.72
22.66
23.62
24.60
17.39
18.23
19.10
19.98
20.88
21.81
22.75
23.72
24.70
17.47
18.32
19.18
20.07
20.98
21.90
22.85
23.81
24.80
17.56
18.40
19.27
20.16
21.07
22.00
22.94
23.91
24.90
5.0
25.00
25.10
25.20
25.30
25.40
25.50
25.60
25.70
25.81
25.91
Squares of Numbers
10'
[Moving the decimal point one place in N requires a corresponding move of two
places in N 2 ]
I
F o
, .
3
4
5
6
7
8
9
5.0
25.00
25.10 j 25.20
25.30
25.40
25.50
25.60
25.70
25.81
25.91
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
26.01
27.04
28.09
29.16
30.25
31.36
32.49
33.64
34.81
26.11
27.14
28.20
29.27
30.36
31.47
32.60
33.76
34.93
26.21
27.25
28.30
29.38
30.47
31.58
32.72
33.87
35.05
26.32
27.35
28.41
29.48
30.58
31.70
32.83
33.99
35.16
26.42
27.46
28.52
29.59
30.69
31.81
32.95
34.11
35.28
26.52
27.56
28.62
29.70
30.80
31.92
33.06
34.22
35.40
26.63
27.67
28.73
29.81
30.91
32.04
33.18
34.34
35.52
26.73
27.77
28.84
29.92
31.02
32.15
33.29
34.46
35.64
26.83
27.88
28.94
30.03
31.14
32.26
33.41
34.57
35.76
26.94
27.98
29.05
30.14
31.25
32.38
33.52
34.69
35.88
6.0
36.00
36.12
36.24
36.36
36.48
36.60
36.72
36.84
36.97
37.09
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
6.9
37.21
38.44
39.69
40.96
42.25
43.56
44.89
46.24
47.61
37.33
38.56
39.82
41.09
42.38
43.69
45.02
46.38
47.75
37.45
38.69
39.94
41.22
42.51
43.82
45.16
46.51
47.89
37.58
38.81
40.07
41.34
42.64
43.96
45.29
46.65
48.02
37.70
38.94
40.20
41.47
42.77
44.09
45.42
46.79
48.16
37.82
39.06
40.32
41.60
42.90
44.22
45.56
46.92
48.30
37.95
39.19
40.45
41.73
43.03
44.36
45.70
47.06
48.44
38.07
39.31
40.58
41.86
43.16
44.49
45.83
47.20
48.58
38.19
39.44
40.70
41.99
43.30
44.62
45.97
47.33
48.72
38.32
39.56
40.83
42.12
43.43
44.76
46.10
47.47
48.72
7.0
49.00
49.14
49.28
49.42
49.56
49.70
49.84
49.98
50.13
50.27
7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8
7.9
50.41
51.84
53.29
54.76
56.25
57.76
59.29
60.84
62.41
50.55
51.98
53.44
54.91
56.40
57.91
59.44
61.00
62.57
50.69
52.13
53.58
55.06
56.55
58.06
59.60
61.15
62.73
50.84
52.27
53.73
55.20
56.70
58.22
59.75
61.31
62.88
50.98
52.42
53.88
55.35
56.85
58.37
59.91
61.47
63.04
51.12
52.56
54.02
55.50
57.00
58.52
60.06
61.62
63.20
51.27
52.71
54.17
55.65
57.15
58.68
60.22
61.78
63.36
51.41
52.85
54.32
55.80
57.30
58.83
60.37
61.94
63.52
51.55
53.00
54.46
55.95
57.46
58.98
60.53
62.09
63.68
51.70
£3.14
54.61
56.10
57.61
59.14
60.68
62.25
63.84
8.0 64.00
64.16
64.32
64.48 64.64
64.80
64.96
65.12
65.29
65.45
8.1
8.2
8.3
8.4
8.5
8.6
8.7
8.8
8.9
65.61
67.24
68.89
70.56
72.25
73.96
75.69
77.44
79.21
65.77
67.40
69.06
70.73
72.42
74.13
75.86
77.62
79.39
65.93
67.57
69.22
70.90
72.59
74.30
76.04
77.79
79.57
66.10
67.73
69.39
71.06
72.76
74.48
76.21
77.97
79.74
66.26
67.90
69.56
71.23
72.93
74.65
76.39
78.15
79.92
66.42
68.06
69.72
71.40
73.10
74.82
76.56
78.32
80.10
66.59
68.23
69.89
71.57
73.27
75.00
76.74
78.50
80.28
66.75
68.39
70.06
71.74
73.44
75.17
76.91
78.68
80.46
66.91
68.56
70.22
71.91
73.62
75.34
77.08
78.85
80.64
67.08
68.72
70.39
72.08
73.79
75.52
77.26
79.03
80.82
9.0
81.00
81.18
81.36
81.54
81.72
81.90
82.08
82.26
82.45
82.63
9.1
9.2
9.3
9.4
9.5
9.6
9.7
9.8
9.9
82.81
84.64
86.49
88.36
90.25
92.16
94.09
96.04
98.01
82.99
84.82
86.68
88.55
90.44
92.35
94.28
96.24
98.21
83.17
85.00
86.86
88.74
90.63
92.54
94.48
96.43
98.41
83.36
85.19
87.05
88.92
90.82
92.74
94.67
96.63
98.60
83.54
85.38
87.24
89.11
91.01
92.93
94.87
96.83
98.80
83.72
85.56
87.42
89.30
91.20
93.12
95.06
97.02
99.00
83.91
85.75
87.61
89.49
91.39
93.32
95.26
97.22
99.20
84.09
85.93
87.80
89.68
91.58
93.51
95.45
97.42
99.40
84.27
86.12
87.99
89.87
91.78
93.70
95.65
97.61
99.60
84.46
86.30
88.17
90.06
91.97
93.90
95.84
97.81
99.80
108
Powers and Roots
Squares and Cubes Square Roots and Cube Roots
No.
Square
Cube
Square
Eoot
Cube
Root
No.
Square
Cube
Square
Root
Cube
Root
1
1
1
1.000
1.000
51
2,601
132,651
7.141
3.708
2
4
8
1.414
1.260
52
2,704
140,608
7.211
3.733
3
9
27
1.732
1.442
53
2,809
148,877
7.280
3.756
4
16
64
2.000
1.587
54
2,916
157,464
7.348
3.780
5
25
125
2.236
1.710
55
3,025
166,375
7.416
3.803
6
36
216
2.449
1.817
56
3,136
175,616
7.483
3.826
7
49
343
2.646
1.913
57
3,249
185,193
7.550
3.849
8
64
512
2.828
2.000
58
3,364
195,112
7.616
3.871
9
81
729
3.000
2.080
59
3,481
205,379
7.681
3.893
10
100
1,000
3.162
2.154
60
3,600
216,000
7.746
3.915
11
121
1,331
3.317
2.224
61
3,721
226,981
7.810
3.936
12
144
1,728
3.464
•2.289
62
3,844
238,328
7.874
3.958
13
169
2,197
3.606
2.351
63
3,969
250,047
7.937
3.979
14
196
2,744
3.742
2.410
64
4,09(5
262,144
8.000
4.000
15
225
3,375
3.873
2.466
65
4,225
274,625
8.062
4.021
16
256
4,096
4.000
2.520
66
4,356
287,496
8.124
4.041
17
289
4,913
4.123
2.571
67
4,489
300,763
8.185
4.0(32
18
324
5,832
4.243
2.621
68
4,624
314,432
8.246
4.082
19
361
6,859
4.359
2.668
69
4,761
328,509
8.307
4.102
20
400
8,000
4.472
2.714
70
4,900
343,000
8.367
4.121
21
441
9,261
4.583
2.759
71
5,041
357.911
8.426
4.141
22
484
10,648
4.690
2.802
72
5,184
373,248
8.485
4.1(50
23
529
12,167
4.796
2.844
73
5,329
389,017
8.544
4.179
24
576
13,824
4.899
2.884
74
5,476
405,224
8.602
4.198
25
625
15,625
5.000
2.924
75
5,625
421,875
8.660
4.217
26
676
17,576
5.099
2.962
76
5,776
438,976
8.718
4.236
27
729
19,683
5.196
3.000
77
5,929
456,533
8.775
4.254
28
784
21,952
5.292
3.037
78
6,084
474,552
8.832
4.273
29
841
24,389
5.385
3.072
79
6,241
493,039
8.888
4.291
30
900
27,000
5.477
3.107
80
6,400
512,000
8.944
4.309
31
961
29,791
5.568
3.141
81
6,561
531,441
9.000
4.327
32
1,024
32,768
5.657
3.175
82
6,724
551,368
9.055
4.344
33
1,089
35,937
5.745
3.208
83
6,889
571,787
9.110
4.362
34
1,156
39,304
5.831
3.240
84
7,056
592,704
9.165
4.380
35
1,225
42,875
5.916
3.271
85
7,225
614,125
9.220
4.397
36
1,296
46,656
6.000
3.302
86
7,396
636,056
9.274
4.414
37
1,369
50,653
6.083
3.332
87
7,569
658,503
9.327
4.431
38
1,444
54,872
6.164
3.362
88
7,744
681,472
9.381
4.448
39
1,521
59,319
6.245
3.391
89
7,921
704,969
9.434
4.465
40
1,600
64,000
6.325
3.420
90
8,100
729,000
9.487
4.481
41
1,681
68,921
6.403
3.448
91
8,281
753,571
9.539
4.498
42
1,764
74,088
6.481
3.476
92
8,464
778,688
9.592
4.514
43
1,849
79,507
6.557
3.503
93
8,649
804,357
9.644
4.531
44
1,936
85,184
6.633
3.530
94
8,836
830,584
9.695
4.547
45
2,025
91,125
6.708
3.557
95
9,025
857,375
9.747
4.563
46
2,116
97,336
6.782
3.583
96
9,216
884,736
9.798
4.579
47
2,209
103,823
6.856
3.609
97
9,409
912,673
9.849
4.595
48
2,304
110,592
6.928
3.634
98
9,604
941,192
9.899
4.610
49
2,401
117,649
7.000
3.659
99
9,801
970,299
9.950
4.626
50
2,500
125,000
7.071
3.684
100
10,000
1,000,000
10.000
4.642
For a more complete table, see The Macjuillan Tables, pp. 94111.
Important Constants
109
Certain Convenient Values for n = 1 to n = 10
n
1/n
Vn
■y/n
n\
1/nl
Logio 11
1
1.000000
1.00000
1.00000
1
1.0000000
0.000000000
2
0500000
1.41421
1.25992
2
0.5000000
0.301029996
3
0.333333
1.73205
1.44225
6
0.1666667
0.477121255
4
0.250000
2.00000
1.58740
24
0.0416667
0.602059991
5
0.200000
2.23607
1.70998
120
0.0083333
0.698970004
6
0.166667
2.44949
1.81712
720
0.0013889
0.778151250
7
0.142857
2.64575
1.91293
5040
0.0001984
0.845098040
8
0.125000
2.82843
2.00000
40320
0.0000248
0.903089987
9
0.111111
3.00000
2.08008
362880
0.0000028
0.954242509
10
0.100000
3.16228
2.15443
3628800
0.0000003
1.000000000
Logarithms of Important Constants
71 =■ NUMBER
Value of n
Log io n
IT
3.14159265
0.49714987
14 7T
0.31830989
9.50285013
7r2
9.86960440
0.99429975
VtF
1.77245385
0.24857494
e = Napierian Base
2.71828183
0.43429448
M= logw e
0.43429448
9.63778431
l5if=log e 10
2.30258509
0.36221569
180 7 7r = degrees in 1 radian
57.2957795
1.75812262
7r r 180 = radians in 1°
0.01745329
8.24187738
ir 4 10800 = radians in 1'
0.0002908882
6.46372613
t 7 648000 = radians in 1"
0.000004848136811095
4.68557487
sin 1"
0.000004848136811076
4.68557487
tan 1"
0.000004848136811152
4.68557487
centimeters in 1 ft.
30.480
1.4840158
feet in 1 cm.
0.032808
8.5159842
inches in 1 m.
39.37 (exact legal value)
1.5951654
pounds in 1 kg.
2.20462
0.3433340
kilograms in 1 lb.
0.453593
9.6566660
g (average value)
32.16 ft./sec./sec.
1.5073
= 981 cm./sec/sec
2.9916690
weight of 1 cu. ft. of water
62.425 lb. (max. density)
1.7953586
weight of 1 cu. ft. of air
0.0807 lb. (at 32° F.)
8.907
cu. in. in 1 (U. S.) gallon
231 (exact legal value)
2.3636120
ft. lb. per sec. in 1 H. P.
550. (exact legal value)
2.7403627
kg. m. per sec. in 1 H. P.
76.0404
1.8810445
watts in 1 H. P.
745.957
2.8727135
11C
1
Fo
ur ]
*lac
e L(
)gar
ithr
US
N
1
2
3
4
5
6
7
8
9
12 3
4 5 6
7 8 9
10
0000
0043
0080
0128
0170
0212
0253
0294
0334
0374
4 8 12
17 21 25
29 33 37
11
12
13
14
15
16
17
18
19
0414
0792
1139
1461
1761
2041
2304
2553
2788
0453
0828
1173
1492
1790
2068
2330
2577
2810
0492
0864
1206
1523
1818
2095
2355
2601
2833
0531
0899
1239
1553
1847
2122
2380
2625
2856
0569
0934
1271
1584
1875
2148
2405
2648
2878
0607
0969
*1303
1614
1903
2175
2430
2672
2900
0645
1004
1335
1644
1931
2201
2455
2695
2923
0682
1038
1367
1673
1959
2227
2480
2718
2945
0719
1072
1399
1703
1987
2253
2504
2742
2967
0755
1106
1430
1732
2014
2279
2529
2765
2989
4 8 11
3 7 10
3 6 10
3 6 9
3 6 8
3 5 8
2 5 7
2 5 7
2 4 7
15 If 23
14 17 21
13 16 19
12 15 18
11 14 17
11 13 16
10 12 15
9 12 14
9 11 13
26 30 34
24 28 31
23 26 29
21 24 27
20 22 25
18 21 24
17 20 22
16 19 21
16 18 20
20
3010
3032
3054
3075
3096
3118
3139
3160
3181
3201
2 4 6
8 1113
15 17 19
21
22
23
24
25
26
27
1
3222
3424
3617
3802
3979
4150
4314
4472
4624
3243
3444
3636
3820
3997
4166
4330
4487
4639
3263
3464
3655
3838
4014
4183
4346
4502
4654
3284
3483
3674
3856
4031
4200
4362
4518
4669
3304
3502
3692
3874
4048
4216
4378
4533
4683
3324
3522
3711
3892
4065
4232
4393
4548
4698
3345
3541
3729
3909
4082
.4249
4409
4564
4713
3365
3560
3747
3927
4099
4265
4425
4579
4728
3385
3579
3766
3945
4110
4281
4440
4594
4742
3404
3598
3784
3962
4133
4298
4456
4609
4757
2 4 6
2 4 6
2 4 6
2 4 5
2 4 5
2 3 5
2 3 5
2 3 5
13 4
8 10 12
8 10 12
7 9 11
7 9 11
7 9 10
7 8 10
6 8 9
6 8 9
6 7 9
14 16 18
14 16 17
13 15 17
12 14 16
12 14 16
11 13 15
11 12 14
11 12 14
10 12 13
30
31
32
33
34
35
36
37
38
39
4771
4786
4928
5065
5198
5328
5453
5575
5694
5809
5922
4800
4814
4829
4843
4857
4871
4886
4900
13 4
6 7 9
10 11 13
4914
5051
5185
5315
5441
5563
5682
5798
5911
4942
5079
5211
5340
5405
5587
5705
5821
5933
4955
5092
5224
5353
5478
5599
5717
5832
5944
4969
5105
5237
5366
5490
5611
5729
5843
5955
4983
5119
5250
5378
5502
5623
5740
5855
5966
4997
5132
5263
5391
5514
5635
5752
5866
5977
5011
5145
5276
5403
5527
5647
5763
5877
5988
5024
5159
5289
5416
5539
5658
5775
5888
5999
5038
5172
5302
5428
5551
5670
5786
5899
6010
13 4
13 4
13 4
12 4
12 4
12 4
12 4
1 2 3
1 2 3
5 7 8
5 7 8
5 7 8
5 6 8
5 6 7
5 6 7
5 6 7
5 6 7
4 5 7
10 11 12
91112
9 1112
9 10 11
9 10 11
8 1011
8 911
8 9 10
8 910
40
6021
6031
6042
6053
6064
6075
6085
6096
6107
6117
12 3
4 5 6
8 9 10
41
42
43
44
45
46
47
48
49
6128
6232
6335
6435
6532
6628
6721
6812
6902
6138
6213
6345
6444
6542
6637
6730
6821
6911
6149
6253
6355
6454
6551
6646
6739
6830
6920
6160
6263
6365
6464
6561
6656
6749
6839
6928
6170
6274
6375
6474
6571
6665
6758
6848
6937
6180
6284
6385
6484
6580
6675
6767
6857
6946
6191
6294
6395
6493
6590
6684
6776
6866
6955
6201
6304
6405
6503
6599
6693
6785
6875
6964
6212
6314
6415
6513
6609
6702
6794
6884
6972
6222
6325
6425
6522
6618
6712
6803
6893
6981
12 3
12 3
12 3
12 3
12 3
12 3
12 3
12 3
12 3
4 5 6
4 5 6
4 5 6
4 5 6
4 5 6
4 5 6
4 5 6
4 5 6
4 4 5
7 8 9
7 8 9
7 8 9
7 8 9
7 8 9
7 7 8
7 7 8
7 7 8
6 7 8
50
6990
6998
7007
7016
7024
7033
7042
7050
7059
7067
12 3
3 4 5
6 7 8
51
52
53
54
7076
7160
7243
7324
7084
7168
7251
7332
7093
7177
7259
7340
7101
7185
7267
7348
7110
7193
7275
7356
7118
7202
7284
7364
7126
7210
7292
7372
7135
7218
7300
7380
7143
7226
7308
7388
7152
7235
7316
7396
12 3
12 3
12 2
12 2
3 4 5
3 4 5
3 4 5
3 4 5
6 7 8
6 7 7
6 6 7
6 6 7
I
1
2
3
4
5
6
7
8
9
12 2
4 5 6
7 8 9
The proportional parts are stated in fall for every tenth at the righthand side.
The logarithm of any number of four significant figures can be read directly by add
Four Place Logarithms
111
If
1
2
3
4
5
6
7
8
9
12 3
4 5 6
7 8 9
55
56
57
58
59
7404
7482
7559
7634
7709
7412
7490
7566
7642
7716
7419
7497
7574
7649
7723
7427
7505
7582
7657
7731
7435
7513
7589
7664
7738
7443
7520
7597
7672
7745
7451
7528
7604
7679
7752
7459
7536
7612
7686
7760
7466
7543
7619
7694
7767
7474
7551
7627
7704
7774
12 2
12 2
1 1 2
1 1 2
112
3 4 5
34 5
3 4 5
3 4 4
3 4 4
5 6 7
5 6 7
5 6 7
5 6 7
5 6 7
60
7782
7789
7796
7803
7810
7818
7825
7832
7839
7846
112
3 4 4
5 6 6
61
62
63
64
65
66
67
68
69
7853
7924
7993
8062
8129
8195
8261
8325
8388
7860
7931
8000
8069
8136
8202
8267
8331
8398
7868
e
8075
8142
8209
8274
83:58
8401
7875
7945
8014
8082
8149
8215
8280
8344
8407
7882
7952
8021
8089
8156
8222
8287
8351
8414
7889
7959
8028
8096
8162
8228
8293
8357
8420
7896
7966
8035
8102
8169
8235
8299
8363
8426
7903 7910 7917
7973 7980 7987
8041 8048 8055
8109 8116 8122
8176 8182 8189
8241 8248 8254
8306 8312 8319
8370 8376 8382
8432 8439 8445
112
1 1 2
112
112
112
112
112
112
112
3 3 4
3 3 4
3 3 4
3 3 4
3 3 4
3 3 4
3 3 4
3 3 4
3 3 4
5 6 6
5 5 6
5 5 6
5 5 6
5 5 6
5 5 6
5 5 6
4 5 6
4 5 6
70
8451
8457
8463
8470
8476
8482
8488
8494
8500 8506
112
3 3 4
4 5 6
71
72
73
74
75
76
77
78
79
8513
8573
8633
8692
8751
8808
8865
8921
8976
8519
8579
8639
8698
8756
8814
8871
8927
8982
8525
8585
8645
8704
8762
8820
8876
8932
8987
8531
8591
8651
8710
8768
8825
8882
8938
8993
8537
8597
8657
8716
8774
8831
8887
8943
8998
8543
8603
8663
8722
8779
8837
8893
8949
9004
8549
8609
8669
8727
8785
8842
8899
8954
9009
8555
8615
8675
87&3
8791
8848
8904
8960
9015
8561
8621
8681
8739
8797
8854
8910
8965
9020
8567
8627
8686
8745
8802
8859
8915
8971
9025
112
112
112
112
112
112
1 1 2
112
112
3 3 4
3 3 4
2 3 4
2 3 4
2 3 3
2 3 3
2 3 3
2 3 3
2 3 3
4 5 6
4 5 6
4 5 5
4 5 5
4 5 5
4 4 5
4 4 5
4 4 5
4 4 5
80
9031
9036
9042
9047
9053
9058
9063
9069 9074
9079
1 1 2
2 3 3
4 4 5
81
82
83
84
85
86,
87
88
89
90&5
9138
9191
9243
9294
9345
9395
9445
9494
9090
9143
9196
9248
9299
9350
9400
9450
9499
9096
9149
9201
9253
9304
9355
9405
9455
9504
9101
9154
9206
9258
9309
9360
9410
9460
9509
910(5
9159
9212
9263
9315
9365
9415
9465
9513
9112
9165
9217
9269
9320
9370
9420
9469
9518
9117
9170
9222
9274
9325
9375
9425
9474
9523
9122
9175
9227
9279
9330
9380
9430
9479
9528
9128
9180
9232
9284
93,35
9385
9435
9484
9533
9133
9186
9238
9289
9340
9390
9440
9489
9538
112
112
112
112
112
112
112
Oil
1 1
2 3 3
2 3 3
2 3 3
2 3 3
2 3 3
2 3 3
2 3 3
2 2 3
2 2 3
4 4 5
4 4 5
4 4 5
4 4 5
4 4 5
4 4 5
4 4 5
3 4 4
3 4 4
90
9542
9547
9552
9557
9562
9566
9571
9576
9581
958(5
Oil
2 2 3
3 4 4
91
92
93
94
95
96
97
98
99
9590
9638
9685
9731
9777
9823
9868
9912
9956
9595
9643
9689
9736
9782
9827
9872
9917
9961
9600
9647
9694
9741
9786
9832
9877
9921
9965
9605
9652
9699
9745
9791
9836
9881
9926
9969
9609
9657
9703
9750
9795
9841
9886
993(1
9974
9614
9661
9708
9754
9800
9845
9890
9934
9978
9619
9666
9713
9759
9805
9850
9894
9939
9983
9624
9671
9717
9763
9809
9854
9899
9943
9987
9628
9675
9722
9768
9814
9859
9903
9948
9991
9633
9680
9727
9773
9818
9863
9908
9952
9996
1 1
Oil
Oil
Oil
Oil
Oil
1 1
Oil
1 1
2 2 3
2 2 3
2 2 3
2 2 3
2 2 3
2 2 3
2 2 3
2 2 3
2 2 3
3 4 4
3 4 4
3 4 4
3 4 4
3 4 4
3 4 4
3 4 4
3 3 4
3 3 4
N
1
2
3
4
5
6
7
8
9
12 3
4 5 6
7 8 9
ing the proportional part corresponding to the fourth figure to the tabular number
corresponding to the first three figures. There may be an error of 1 in the last place.
112
Four Place Trigonometric Functions
[Characteristics of Logarithms omitted —
determine by the usual rule from the value]
Radians
Degbees
Sine
Tangent
Cotangent
Cosine
Value
Log 10
Value Log 10
Value
Logio
Value
Log 10
.0000
.0029
0°00'
10
.0000
.0029
.0000
.0029 .4637
1.0000
.0000
90° 00'
50
1.5708
1.5679
.4637
343.77
.5363
i!oooo
!oooo
.0058
20
.0058
.7648
.0058 .7648
171.89
.2352
1.0000
.0000
40
1.5650
.0087
30
.0087
.9408
.0087 .9409
114.59
.0591
1.0000
.0000
3Q
1.5621
.0116
40
.0116
.0658
.0116 .0658
85.940
.9342
.9999
.0000
20
1.5592
.0145
50
.0145
.1627
.0145 .1627
68.750
.8373
.9999
.0000
10
1.5563
.0175
1°00'
.0175
.2419
.0175 .2419
57.290
.7581
.9998
.9999
89° 00'
1.5533
.0204
10
.0204
.3088
.0204 .3089
49.104
.6911
.9998
.9999
50
1.5504
.0233
20
.0233
.3668
.0233 .3669
42.964
.6331
.9997
.9999
40
1.5475
.0262
30
.0262
.4179
.0262 .4181
38.188
.5819
.9997
.9999
30
1.5446
.0291
40
.0291
.4637
.0291 .4638
34.368
.5362
.9996
.9998
20
1.5417
.0320
50
.0320
.5050
.0320 .5053
31.242
.4947
.9995
.9998
10
1.5388
.0349
2° 00'
.0349
.5428
.0349 .5431
28.636
.4569
.9994
.9997
88° 00'
1.5359
.0378
10
.0378
.5776
.0378 .5779
26.432
.4221
.9993
.9997
50
1.5330
.0407
20
.0407
.6097
.0407 .6101
24.542
.3899
.9992
.9996
40
1.5301
.0436
30
.0436
.6397
.0437 .6401
22.904
.3599
.9990
.9996
30
1.5272
.0465
40
.0465
.6677
.0466 .6682
21.470
.3318
.9989
.9995
20
1.5243
.0495
50
.0494
.6940
.0495 .6945
20.20(3
.3055
.9988
.9995
10
1.5213
.0524
3° 00'
.0523
.7188
.0524 .7194
19.081
.2806
.9986
.9994
87° 00'
1.5184
.0553
10
.0552
.7423
.0553 .7429
18.075
.2571
.9985
.9993
50
1.5155
.0582
20
.0581
.7645
.0582 .7652
17.169
.2348
.9983
.9993
40
1.5126
.0611
30
.0610
.7857
.0612 .7865
16.350
.2135
.9981
.9992
30
1.5097
.0640
40
.0640
.8059
.0641 .8067
15.605
.1933
.9980
.9991
20
1.5068
.0669
50
.0669
.8251
.0670 .8261
14.924
.1739
.9978
.9990
10
1.5039
.0698
4° 00'
.0698
.8436
.0699 .8446
14.301
.1554
.9976
.9989
86° 00'
1.5010
.0727
10
.0727
.8613
.0729 .8624
13.727
.1376
.9974
.9989
50
1.4981
.0756
20
.0756
.8783
.0758 .8795
13.197
.1205
.9971
.9988
40
1.4952
.0785
30
.0785
.8946
.0787 .8960
12.706
.1040
.9969
.9987
30
1.4923
.0814
40
.0814
.9104
.0816 .9118
12.251
.0882
.9967
.9986
20
1.4893
.0844
50
.0843
.9256
.0846 .9272
11.826
.0728
.9964
.9985
10
1.4864
.0873
5° 00'
.0872
.9403
.0875 .9420
11.430
.0580
.9962
.9983
85° 00'
1.4835
.0902
10
.0901
.9545
.0904 .9563
11.059
.0437
.9959
.9982
50
1.4806
.0931
20
.0929
.9682
.0934 .9701
10.712
.0299
.9957
.9981
40
1.4777
.0960
30
.0958
.9816
.0963 .9836
10.385
.0164
.9954
.9980
30
1.4748
.0989
40
.0987
.9945
.0992 .9966
10.078
.0034
.9951
.9979
20
1.4719
.1018
50
.1016
.0070
.1022 .0093
9.7882
.9907
.9948
.9977
10
1.4690
.1047
6° 00'
.1045
.0192
.1051 .0216
9.5144
.9784
.9945
.9976
84° 00'
1.4661
.1076
10
.1074
.0311
.1080 .0336
9.2553
.9664
.9942
.9975
50
1.4632
.1105
20
.1103
.0426
.1110 .0453
9.0098
.9547
.9939
.9973
40
1.4603
.1134
30
.1132
.0539
.1139 .0567
8.7769
.9433
.9936
.9972
30
1.4573
.1164
40
.1161
.0648
.1169 .0678
8.5555
.9322
.9932
.9971
20
1.4544
.1193
50
.1190
.0755
.1198 .0786
8.3450
.9214
.9929
.9969
10
1.4515
.1222
7° 00'
.1219
.0859
.1228 .0891
8.1443
.9109
.9925
.9968
83° 00'
1.4486
.1251
10
.1248
.0961
.1257 .0995
7.9530
.9005
.9922
.9966
/50
1.4457
.1280
20
.1276
.1060
.1287 .1096
7.7704
.8904
.9918
.9964
r40
*30
1.4428
.1309
30
.1305
.1157
.1317 .1194
7.5958
.8806
.9914
.9963
1.4399
.1338
40
.1334
.1252
.1346 .1291
7.4287
.8709
.9911
.9961
20
1.4370
.1367
50
.1363
.1345
.1376 .1385
7.2687
.8615
.9907
.9959
10
1.4341
.1396
8° 00'
.1392
.1436
.1405 .1478
7.1154
.8522
.9903
.9958
82° 00'
1.4312
.1425
10
.1421
.1525
.1435 .1569
6.9682
.8431
.9899
.9956
50
1.4283
.1454
20
.1449
.1612
.1465 .1658
6.8269
.8342
.9894
.9954
40
1.4254
.1484
30
.1478
.1697
.1495 .1745
6.6912
.8255
.9890
.9952
30
1.4224
.1513
40
.1507
.1781
.1524 .1831
6.5606
.8169
.988(5
.9950
20
1.4195
.1542
50
.1536
.1863
.1554 .1915
6.4348
.8085
.9881
.9948
10
1.4166
.1571
9° 00'
.1564
.1943
.1584 .1997
6.3138
.8003
.9877
.9946
81° 00'
1.4137
Value
Log 10
Value Lojr 10
Value
Log 10
Value
Log 10
Degrees
Radians
Cosine
Cotangent
Tangent
Sine
Four Place Trigonometric Functions
113
[Characteristics of Logarithms omitted —
ietermine by the usual rule from the value]
Radians
Degbees
Sine
Tangent
Cotangent
Cosine
Value
Log 10
Value
Logio
Value Log 10
Value
L«g 10
.1571
9° 00'
.1564
.1943
.1584
.1997
6.3138 .8003
.9877
.9946
81° 00'
1.4137
.1600
10
.1593
.2022
.1614
.2078
6.1970 .7922
.9872
.9944
50
1.4108
.1629
20
.1622
.2100
.1644
.2158
6.0844 .7842
.9868
.9942
40
1.4079
.1658
30
.1650
.2176
.1673
.2236
5.9758 .7764
L9868
'.9858
.9940
30
1.4050
.1687
40
.1679
.2251
.1703
.2313
5.8708 .7687
.9938
20
1.4021
.1716
50
.1708
.2324
.1733
.2389
5.7694 .7611
.9853
.9936
10
1.3992
.1745
10° 00
.1736
.2397
.1763
.2463
5.6713 .7537
.9848
.9934
80° 00'
1.3963
.1774
10
.1765
.2468
.1793
.2536
5.5764 .7464
.9843
.9931
50
1.3934
.1804
20
.1794
.2538
.1823
.2609
5.4845 .7391
.9838
.9929
40
1.3904
.1833
30
.1822
.2606
.1853
.2680
5.3955 .7320
.9833
.9927
30
1.3875
.1862
40
.1851
.2674
.1883
.2750
5.3093 .7250
.9827
.9924
20
1.3846
.1891
50
.1880
.2740
.1914
.2819
5.2257 .7181
.9822
.9922
10
1.3817
.1920
11°00'
.1908
.2806
.1944
.2887
,5.1446 .7113
.9816
.9919
79° 00
1.3788
.1949
10
.1937
.2870
:1974
.2953
5.0658 .7047
.9811
.9917
50
1.3759
.1978
20
.1965
.2934
.2004
.3020
4.9894 .6980
.9805
.9914
40
1.3730
.2007
30
.1994
.2997
.2035
.3085
4.9152 .6915
.9799
.9912
30
1.3701
.2036
4D
.2022
.3058
.2065
.3149
4.8430 .6851
.9793
.9909
20
1.3672
.2065
50
.2051
.3119
.2095
.3212
4.7729 .6788
.9787
.9907
10
1.3643
.2094
12° 00'
.2079
.3179
.2126
.3275
4.7046 .6725
.9781
.9904
78° 00'
L3614
.2123
10
.2108
.3238
.2156
.3336
4.6382 .6664
.9775
.9901
50
1.3584
.2153
20
.2136
.3296
.2186
.3397
4.5736 .6603
.9769
.9899
40
1.3555
.2182
30
.2164
.3353
.2217
.3458
•4.5107 .6542
.9763
.9896
30
1.3526
1 .2211
40
.2193
.3410
.2247
.3517
4.4494 .6483
.9757
.9893
20
1.3497
.2240
50
.2221
.3466
.2278
.3576
4.3897 .6424
.9750
.9890
10
1.3468
.2269
13° 00'
.2250
.3521
.2309
.3634
4.3315 .6366
.9744
.9887
77° 00'
1.3439
.2298
10
.2278
.3575
.2339
.3691
4.2747 .6309
.9737
.9884
50
1.3410
.2327
20
.2306
.3629
.2370
.3748
4.2193 .6252
.9730
.9881
40
1.3381
.2356
30
.2334
.3682
.2401
.3804
4.1653 .6196
.9724
.9878
30
1.3352
.2385
40
.2363
.3734
.2432
.3859
4.1126 .6141
.9717
.9875
20
1.3323
.2414
50
.2391
.3786
.2462
.3914
4.0611 .6086
.9710
.9872
10
1.3294
.2443
14° 00'
.2419
.3837
.2493
.3968
4.0108 .6032
.9703
.9869
76° 00'
1.3265
.2473
10
.2447
.3887
.2524
.4021
3.9617 .5979
.9696
.9866
50
1.3235
.2502
20
.2476
.3937
.2555
.4074
3.9136 .5926
.9689
.9863
40
1.3206
.2531
30
.2504
.3986
.2586
.4127
3.8667 .5873
.9681
.9859
30
1.3177
.2560
40
.2532
.4035
.2617
.4178
3.8208 .5822
.9674
.9856
20
1.3148
.2589
50
.2560
.4083
.2648
.4230
3.7760 .5770
.9667
.9853
10
1.3119
.2618
15°00'
.2588
.4130
.2679
.4281
3.7321 .5719
.9659
.9849
75° 00'
1.3090
.2647
10
.2616
.4177
.2711
.4331
3.6891 .5669
.9652
.9846
50
1.3061
.2676
20
.2644
.4223
.2742
.4381
3.6470 .5619
.9644
.9843
40
1.3032
.2705
30
.2672
.4269
.2773
.4430
3.6059 .5570
.9636
.9839
30
1.3003
.2734
40
.2700
.4314
.2805
.4479
3.5656 .5521
.9628
.9836
20
1.2974
.2763
50
.2728
.4359
.2836
.4527
3.5261 .5473
.9621
.9832
10
1.2945
.2793
16° 00'
.2756
.4403
.2867
.4575
3.4874 .5425
.9613
.9828
74° 00'
1.2915
.2822
10
.2784
.4447
.2899
.4622
3.4495 .5378
.9605
.9825
50
1.2886
.2851
20
.2812
.4491
.2931
.4669
3.4124 .5331
.9596
.9821
40
1.2857
.2880
30
.2840
.4533
.2962
.4716
3.3759 .5284
.9588
.9817
30
1.2828
.2909
40
.2868
.4576
.2994
.4762
3.3402 .5238
.9580
.9814
20
1.2799
.2938
50
.2896
.4618
.3026
.4808
3.3052 .5192
.9572
.9810
10
1.2770
.2967
17° 00'
.2924
.4659
.3057
.4853
3.2709 .5147
.9563
.9806
73° 00'
1.2741
.2996
10
.2952
.4700
.3089
.4898
3.2371 .5102
.9555
.9802
50
1.2712
.3025
20
.2979
.4741
.3121
.4943
3.2041 .5057
.9546
.9798
40
1.2683
.3054
30
.3007
.4781
.3153
.4987
3.1716 .5013
.9537
.9794
30
1.2654
.3083
40
.3035
.4821
.3185
.5031
3.1397 .4969
.9528
.9790
20
1.2625
.3113
50
.3062
.4861
.3217
.5075
3.1084 .4925
.9520
.9786
10
1.2595
.3142
18° 00'
.3090
.4900
.3249
.5118
3.0777 .4882
.9511
.9782
72° 00'
1.2566
Value
Logio
Value
Log 10
Value Log 10
Value
Log 10
Degrees
Radians
Cosine
Cotangent
Tangent
Sine
114
Four Place Trigonometric Functions
[Characteristics of Logarith
ms omitted —
determine by the usual rule from the value]'
Radians
Degrees
Sine
Tangent
Cotangent
Cosine
Value
L°g 10
Value
Log x
Value
Log 10
Value Log 10
.3142
18° 00'
.3090
.4900
.3249
.5118
3.0777
.4882
.9511 .9782
72° 00'
1.2566
.3171
10
.3118
.4939
.3281
.5161
3.0475
.4839
.9502 .9778
50
1 .2537
.3200
20
.3145
.4977
.3314
.5203
3.0178
.4797
.9492 .9774
40
1.2508
.3229
30
.3173
.5015
.3346
.5245
2.9887
.4755
.9483 .9770
30
1.2479
.3258
40
.3201
.5052
.3378
.5287
2.9600
.4713
.9474 .9765
20
1.2450
.3287
50
.3228
.5090
.3411
.5329
2.9319
.4671
.9465 .9761
10
1.2421
.3316
19° 00'
.3256
.5126
.3443
.5370
2.9042
.4630
.9455 .9757
71° 00'
1.2392
.3345
10
.3283
.5163
.3476
.5411
2.8770
.4589
.9446 .9752
50
1.2363
.3374
20
.3311
.5199
.3508
.5451
2.8502
.4549
.9436 .9748
40
1.2334
.3403
30
.3338
.5235
.3541
.5491
2.8239
.4509
.9426 .9743
30
1.2305
.3432
40
.3365
.5270
.3574
.5531
2.7980
.4469
.9417 .9739
20
1.2275
.3462
50
.3393
.5306
.3607
.5571
2.7725
.4429
.9407 .9734
10
1.2246
.3491
20° 00'
.3420
.5341
.3640
.5611
2.7475
.4389
.9397 !9730
70° 00'
1.2217
.3520
10
.3448
.5375
.3073
.5650
2.7228
.4350
.9387 .9725
50
1.2188
.3549
20
.3475
.5409
.3706
.5689
2.6985
.4311
.9377 .9721
40
1.2159
.3578
30
.3502
.5443
.3739
.5727
2.6746
.4273
.9367 .9716
30
1.2130
.3607
40
.3529
.5477
.3772
.5766
2.6511
.4234
.9356 .9711
20
1.2101
.3636
50
.3557
.5510
.3805
.5804
2.6279
.4196
.9346 .9706
10
1.2072
.3665
21° 00'
.3584
.5543
.3839
.5842
2.6051
.4158
.9336 ,9702
69° 00'
1.2043
.3694
10
.3611
.5576
.3872
.5879
2.5826
.4121
.9325 .9697
50
1.2014
.3723
20
.3638
.5609
.3906
.5917
2.5605
.4083
.9315 .9692
40
1.1985
.3752
30
.3665
.5641
.3939
.5954
2.5386
.4046
.9304 .9687
30
1.1956
.3782
40
.3692
.5673
.3973
.5991
2.5172
.4009
.9293 .9682
20
1.1926
.3811
50
.3719
.5704
.4006
.6028
2.4960
.3972
.9283 ,.9677
10
1.1897
.3840
22° 00'
.3746
.5736
.4040
.6064
2.4751
.3936
.9272 .9672
68° 00'
1.1868
.3869
10
.3773
.5767
.4074
.6100
12.4545
12.4342
.3900
.9261 .9667
50
1.1839
.3898
20
.3800
.5798
.4108
.6136
.3864
.9250 .9661
40
1.1810
.3927
30
.3827
.5828
.4142
.6172
2.4142
.3828
.9239 .9656
30
1.1781
.3956
40
.3854
.5859
.4176
.6208
2.3945
.3792
.9228 .9651
20
1.1752
.3985
50
.3881
.5889
.4210
.6243
2.3750
.3757
.9216 .9646
10
1.1723
.4014
23° 00'
.,3907
.5919
.4245
.6279
2.3559
.3721
.9205 .9640
67° 00'
1.1694
.4043
10
.3934
.5948
.4279
.6314
2.3369
.3686
.9194 .9635
50
1.1665
.4072
20
.3961
.5978
.4314
.6348
2,3183
.3652
.9182 .9629
40
1.163(5
.4102
30
.3987
.6007
.4348
.6383
2.2998
.3617
.9171 .9624
30
1.1606
.4131
40
.4014
.6036
.4383
.6417
2.2817
.3583
.9159 .9618
20
1.1577
.4160
50
.4041
.6065
.4417
.6452
2.2637
.3548
.9147 .9613
1Q
1.1548
.4189
24° 00'
.4067
.6093
.4452
.6486
2.24(50
.3514
.9135 .9607
66° 00'
1.1519
.4218
10
.4094
.6121
.4487
.6520
2.2286
.3480
.9124 .9602
50
1.1490
.4247
20
.4120
.6149
.4522
.6553
2.2113
.3447
.9112 .9596
40
1.1461
.4276
30
.4147
.6177
.4557
.6587
2.1943
.3413
.9100 .9590
30
1.1432
.4305
40
.4173
.6205
.4592
.6620
2.1775
.3380
.9088 .9584
20
1.1403
.4334
50
.4200
.6232
.4628
.6654
2.1609
.3346
.9075 .9579
10
1.1374
.4363
25° 00'
.4226
.6259
.4663
.6687
2.1445
.3313
.9063 .9573
65° 00'
1.1345
.4392
10
.4253
.6286
.4699
.6720
2.1283
.32&0
.9051 .9567
50
1.1316
.4422
20
.4279
.6313
.4734
.6752
2.1123
.3248
.9038 .9561
40
1.1286
.4451
30
.4305
.6340
.4770
.6785
2.0965
.3215
.9026 .9555
30
1.1257
.4480
40
.4331
.6366
.4806
.6817
2.0809
.3183
.9013 .9549
20
1.1228
.4509
50
.4358
.6392
.4841
.6850
2.0655
.3150
.9001 .9543
10
1.1199
.4538
26° 00'
.4384
.6418
.4877
.6882
2.0503
.3118
.8988 .9537
64° 00'
1.1170
.4567
10
.4410
.6444
.4913
.6914
2.0353
.3086
.8975 .9530
50
1.1141
.4596
20
.4436
.6470
.4950
.6946
2.0204
.3054
.8962 .9524
40
1.1112
.4625
30
.4462
.6495
.4986
.6977
2.0057
.3023
.8949 .9518
30
1.1083
.4654
40
.4488
.6521
.5022
.7009
1.9912
.2991
.893(5 .9512
20
1.1054
.4683
50
.4514
.6546
.5059
.7040
1.9768
.2960
.8923 .9505
10
1.1025
.4712
27° 00'
.4540
.6570
.5095
.7072
1.9626
.2928
.8910 .9499
63° 00'
1.0996
Value
Logio
Value
LOftfl
Value
Loffio
Value ' Log 10
Degrees
Radians
Cosine
Cotangent
. Tanoent . Sine
Four Place Trigonometric Functions
115
[Characteristi
cs of Logarithms omitted — determine by the usual rule from the value]
Radians
Degeees
SlXE
Value Log 10
Tangent Cotangent Cosine
Value Log 10 Value Log 10 i Value Log 10
.4712
27° 00'
.4540 .6570
.5095 .7072
1.9626
.2928
.8910 .9499
63° 00'
1.0996
.4741
10
.4566 .6595
.5132 .7103
1.94S6
.2897
.8897 .9492
50
1.0966
.4771
20
.4592 .6620
.5169 .7134
1.9347
.2866
.8884 .9486
40
1.0937
.4800
30
.4617 .6644
.5206 .7165
1.9210
.2835
.8870 .9479
30
1.0908
.4829
40
.4643 .6668
.5243 .7196
1.9074
.2804
.8857 .9473
20
1.0879
.4858
50
.4669 .6692
.5280 .7226
1.8940
.2774
.8843 .9466
10
1.0850
.4887
28° 00'
.4695 .6716
.5317 .7257
1.8807
.2743
.8829 .9459
62° 00'
1.0821
.4916
10
.4720 .6740
.5354 .7287
1.8676
.2713
.8816 .9453
50
1.0792
.4945
20
.4746 .6763
.5392 .7317
1.8546
.2683
.8802 .9446
40
1.0703
.4974
30
.4772 .6787
.5430 .7348
1.8418
.2652
.8788 .9439
30
1.0734
,5003
40
.4797 .6810
.5167 .7378
1.8291
.2622
.8774 .9432
20
1.0705
.5032
50
.4823 .6833
.5505 .7408
1.8165
.2592
.8760 .9425
10
1.0676
.5061
29° 00'
.4848 .6856
.5543 .7438
1.8040
.2562
.8746 .9418
61° 00'
1.0647
.5091
10
.4874 .6878
.5581 .7467
1.7917
.2533
.8732 .9411
50
1.0617
.5120
20
.4899 .6901
.5619 .7497
1.7796
.2503
.8718 .9404
40
1.0688
.5149
30
.4924 .6923
.5658 .7526
1.7675
.2474
.8704 .9397
30
1.0559
.5178
40
.4950 .6946
.5696 .7556
1.7556
.2444
.8689 .9390
20
1.0530
.5207
50
.4975 .6968
.5735 .7585
1.7437
.2415
.8675 .9383
10
1.0501
.5230
30° 00'
.5000 .6990
.5774 .7614
1.7321
.2386
.8660 .9375
60° 00'
1.0472
.5265
10
.5025 .7012
.5812 .7044
1.7205
.2356
.8646 .9368
50
1.0443
.5294
20
.5050 .7033
^5851 .7673
1.7090
.2327
.8631 .9361
40
1.0414
.5323
30
.5075 .7055
.3890 .7701
1.6977
.2299
.8616 .9353
30
1.0385
.5352
40
.5100 .7076
.5930 .7730
1.6864
.2270
.8601 .9346
20
1.0356
.5381
50
.5125 .7097
.5969 .7759
1.6753
.2241
.8587 .9338
10
1.0327
.5411
31° 00'
.5150 .7118
.6009 .7788
1.6643
.2212
.8572 .9331
59° 00'
1.0297
.5440
10
.5175 .7139
.6048 .7816
1.6534
.2184
.8557 .9323
50
1.0268
.5469
20
.5200 .7160
.6088 .7845
1.6426
.2155
.8542 .9315
40
1.0239
.5498
30
.5225 .7181.
.6128 .7873
1.6319
.2127
.8526 .9308
_30
1.0210
.5527
40
.5250 .7201
.6168 .7902
1.6212
.2098
.8511 .9300
20
1.0181
.5556
10
.5275 .7222
.6208 .7930
1.6107
.2070
.8496 .9292
10
1.0152
.5585
32° 00'
.5299 .7242
.6249 .7958
1.6003
.2042
.8480 .9284
58° 00'
1.0123
.5(314
10
.5324 .7262
.6289 .7986
1.5900
.2014
.8465 .9276
50
1.0094
.5643
20
.5318 .7282
.6330 .8014
1.5798
.1986
.8450 .9268
40
1.0065
.5672
30
.5373 .7302
.6371 .8042
1.5697
.1958
.8434 .9260
30
1.0036
.5701
40
.5398 .7322
.6412 .8070
1.5597
.1930
.8418 .9252
20
1.0007
.5730
50
.5422 .7342
.6453 .8097
1.5497
.1903
.8403 .9244
10
.9977
.5760
'33° 00'
.5446 .7361
.6494 .8125
1.5399
.1875
.8387 .9236
57° 00'
.9948
.5789
10
.5471 .7380
.6536 .8153
1.5301
.1847
.8371 .9228
50
.9919
.5818
20
.5495 .7400
.6577 .8180
1.5204
.1820
1^8355 .9219
«8339 .9211
40
.9890
.5847
30
.5519 .7419
.6619 .8208
1.5108
.1792
30
.9861
.5876
40
.5544 .7438
.6661 .8235
1.5013
.1765
.8323 .9203
20
.9832
.5905
50
.5568 .7457
.6703 .8263
1.4919
.1737
.8307 .9194
10
.9803
.5934
34° 00'
.5592 .7476
.6745 .8290
1.4826
.1710
.8290 .9186
56° 00'
.9774
.5963
10
.5616 .7494
.6787 .8317
1.4733
.1683
.8274 .9177
50
.9745
.5992
20
.5640 .7513
.6830 .8344
1.4641
.1656
.8258' .9169
40
.9716
.6021
30
.5664 .7531
.6873 .8371
1.4550
.1629
.8241 .9160
30
.9687
.6050
40
.5688 .7550
.6916 .8398
1.4460
.1602
.8225 .9151
20
.9657
.6080
50
.5712 .7568
.6959 .8425
1.4370
.1575
.8208 .9142
10
.9628
.6109
35° 00'
.5736 .7586
.7002 .8452
1.4281
.1548
.8192 .9134
55° 00'
.9599
.6138
10
.5760 .7604
.7046 .8479
1.4193
.1521
.8175 .9125
50
.9570
.6167
20
.5783 .7622
.7089 .8506
1.4106
.1494
.8158 .9116
40
.9541
.6196
30
.5807 .7640
.7133 .8533
1.4019
.1467
.8141 .9107
30
.9512
.6225
40
.5831 .7657
.7177 .8559
1.3934
.1441
.8124 .9098
20
.9483
.6254
50
.5854 .7675
.7221 .8586
1.3848
.1414
.8107 .9089
10
.9454
.6283
36° 00'
.5878 .7692
.7265 .8613
1.3764
.1387
.8090 .9080
54° 00'
.9425
Value Log 10
Value Loer 10
Value
Log 10
Value Log 10
Degrees
Radians
Cosine
Cotangent
Tangent
Sine
116 Four Place Trigonometric Functions
[Characteristics of Logarithms omitted — determine by the usual rule from the value]
Radians
Degress
Sine
Tangent
Cotangent
Cosine
Value Log 10
Value Log 1(
Value Log 10
Value Log lf
.6283
36° 00'
.5878 .7602
.7265 .8613
1.3704 .1387
.8090 .9080
54° 00'
.9425
.6312
10
.5901 .7710
.7310 .8639
1.3080 .1301
.8073 .9070
50
.9390
.6341
20
.5925 .7727
.7355 .8666
1.3597 .1334
,8050 .9001
40
.9307
.6370
30
.5948 .7744
.7400 .8692
1.3514* .1308
1.3432 7 '.1282
[8039 .9052
30
.9338
.6400
40
.5972 .7761
.7445 .8718
..8021 .9042
20
.9308
.6429
50
.5995 .7778
.7490 .8745
1.3351 .1255
.8004 .9033
10
.9279
.6458
37° 00'
.6018 .7795
.7536 .8771
1.3270 .1229
.7980 .9023
53° 00'
.9250
.6487
10
.6041 .7811
.7581 .8797
1.3190 .1203
.7909 .9014
50
.9221
.6516
20
.6065 .7828
.7627 .8824
1.3111 .1170
17951 .9004
[7934 .8995
40
.9192
.6545
30
.6088 .7844
.7673 .8850
1.3032 .1150
30
.9103
.6574
40
.6111 .7861
.7720 .8870
1.2954 .1124
.7910 .8985
20
.9134
.6603
50
.6134 .7877
.7766 .8902
1.2876 .1098
.7898 .8975
10
.9105
.6632
38° 00'
.6157 .7893
.7813 .8928
1.2799 .1072
.7880 .8905
52° 00'
.9070
.6661
10
.6180 .7910
.7860 .8954
1.27231 .1046
1:2647/ .1020
.7802 .8955
50
.9047
.6690
20
.6202 .7926
,.7907 .8980
'.7954 .9006
.7844 .8945
~40
.9018
.6720
30
.6225 .7941
1.2572 .0994
.7820 .8935
^30
.8988
.6749
.6778
40
50
.6248 .7957
.6271 .7973
,£002 .9032
.8050 .9058
1.24971 .0908
1.24221.0942
.7808 .8925
.7790 .8915
^■20
10
.8959
.8930
.6807
39° 00'
.6293 .7989
.8098 .9084
1.2349 .0910
.7771 .8905
51°00'
.8901
.6836
10
.6316 .8004
.8146 .9110
1.2270 .0890
.7753 .8895
50
.8872
.6865
20
.6338 .8020
.8195 .9135
1.2203 .0805
.7735 .8884
40
.8843
.6894
30
.6361 .8035
.8243 .9161
1.2131 .0839
.7710 .8874
30
.8814
.6923
40
.6383 .8050
.8292 .9187
1.2059 .0813
.7098 .8804
20
.8785
.6952
50
.6406 .8066
.8342 .9212
1.1988 .0788
.7079 .8853
10
.8750
.6981
40° 00'
.6428 .8081
.8391 .9238
1.1918 .0762
.7000 .8843
50° 00'
.8727
.7010
10
.6450 .8096
.8441 .9264
1.1847 .0736
.7042 .8832
50
.8098
.7039
20
.6472 .8111
.8491 .9289
1.1778 .0711
.7023 .8821
40
.8008
.7069
30
.6494 .8125
.8541 .9315
1.1708 .0685
.7604 .8810
30
.8039
.7098
40
.6517 .8140
.8591 .9341
1.1640 .0659
.7585 .8800
20
.8010
.7127
50
.6539 .8155
.8642 .9366
1.1571 .0634
.7566 .8789
10
.8581
.7156
41° 00'
.6561 .8169
.8693 .9392
1.1504 .0608
.7547 .8778
49° 00'
.8552
.7185
10
.6583 .8184
.8744 .9417
1.1436. .0583
.7528 .8767
50
.8523
.7214
20
.6604 .8198
.8796 .9443
1.1369 .0557
.7509 .8756
40
.8494
.7243
30
.6626 .8213
.8847 .9468
1.1303 .0532
1. 1237 .0506
.7490 .8745
30
.8405
.7272
40
.6648 .8227
.8899 .9494
.7470 .8733
20
.8430
.7301
50
.6670 .8241
.8952 .9519
1.1171 .0481
.7451 .8722
10
.8407
.7330
42° 00'
.6691 .8255
.9004 .9544
1.1100 .0450
.7431 .8711
48° 00'
.8378
.7359
10
.6713 .8269
.9057 .9570
1.1041 .0430
.7412 .8699
50
.8348
.7389
20
.6734 .8283
.9110 .9595
1.0977 .0405
.7392 .8688
40
.8319
.7418
30
.6756 .8297
—9163 .9021
.9217 .9(340
1.0913 .0379
.7373 .8676
,30
.8290
.7447
40
.6777 .8311
1.0850. 0354
.7353 .8605
20
.8201
.7476
50
.6799 .8324
.9271 .9071
1.0780 .0329
^7333 .8653
.7314 .8641
10
.8232
.7505
43° 00'
.6820 .8338
.9325 .9097
1.0724 .0303
47° 00'
.8203
.7534
10
.6841 .8351
.9380 .9722
1.0001 .0278
.7294 .8629
50
.8174
.7563
20
.6862 .8365
.9435 .9747
1.0599 .0253
.7274 .8018
40
.8145
.7592
30
.688*^8378
.9490 .9772
1.0538 .0228
1.0477 .0202
.7254 .8000
30
.8110
.7621
40
.6905 .8391
.9545 .9798
.7234 .8594
20
.8087
.7650
50
.6926 .8405
.9001 .9823
1.0410 .0177
.7214 .8582
10
.8058
.7679
44° 00'
.6947 .8418
.9057 .9848
1.0355 .0152
.7193 .8509
46° 00'
.8029
.7709
10
.6967 .8431
.9713 .9874
1.0295 .0120
.7173 .8557
50
.7999
.7738
20
,6988 .8444
.9770 .9899
1.0235 .0101
.7153 .8545
40
.7970
.7767
30
.7009 .8457
.9827 .9924
1.0170 .0070
.7133 .8532
30
.7941
.7796
40
.7030 .8469
.9884 .9949
1.0117 .0051
.7112 .8520
20
.7912
.7825
50
.7050 .8482
.9942 .9975
1.0058 .0025
.7092 .8507
10
.7883
.7854
45° 00'
.7071 .8495
1.0000 .0000
1.0000 .0000
.7071 .8495
45° 00'
.7854
Value Log 10
Value Log 10
Value Log 10
Value Log 10
Degrees
EIadians
Cosine {
Cotangent
Tangent
Sine
Tallies and Logarithms of Haversines
117
[Characteristics of Logarithms omitted 
— determine by rule from the value]
10'
20'
3C
'
40'
. 5C
'
Value
Log 10
Value
Log 10
Value
Log 10
Value
Log 10
Value
Log 10
Value
Log 10
.0000
.0000 4.3254
.0000 4.9275
.0000 5.2796
.0000 5.5295
.0001 5.7233
.0001 5.8817
.0001 6.0156
.0001 6.1315
.0002
.2338
.0002
.3254
.0003
.4081
2
.0003
.4837
.0004
.5532
.0004
.6176
.0005
.6775
.0005
.7336
.0006
.7862
3
.0007
.8358
.0008
.8828
.0008
.9273
.0009
.9697
.0010
.0101
.0011
.0487
4
.0012
.0856
.0013
.1211
.0014
.1551
.0015
.1879
.0017
.2195
.0018
.2499
5
.0019
.2793
.0020
.3078
.0022
.3354
.0023
.3621
.0024
.3880
.0026
.4132
6
.0027
.4376
.0029
.4614
.0031
.4845
.0032
.5071
.0034
.5290
.0036
.5504
7
.0037
.5713
.0039
.5918
.0041
.6117
.0043
.6312
.0045
.6503
.0047
.6689
8
.0049
.6872
.0051
.7051
.0053
.7226
.0055
.7397
.0057
.7566
.0059
.7731
9
.0062
.7893
.0064
.8052
.0066
.8208
.0069
.8361
.0071
.8512
.0073
.8660
10
.0076
.8806
.0079
.8949
.0081
.9090
.0084
.9229
.0086
.9365
.0089
.9499
11
.0092
.9631
.0095
.9762
.0097
.9890
.0100
.0016
.0103
.0141
.0106
.0264
12
.0109
.0385
.0112
.0504
.0115
.0622
.0119
.0738
.0122
.0853
.0125
.0966
13
.0128
.1077
.0131
.1187
.0135
.1296
.0138
.1404
.0142
.1510
.0145
.1614
14
.0149
.1718
.0152
.1820
.0156
.1921
.0159
.2021
.0163
.2120
.0167
.2218
15
.0170
.2314
.0174
.2409
.0178
.2504
.0182
.2597
.0186
.2689
.0190
.2781
16
.0194
.2871
.0198
.2961
.0202
.3049
.0206
.3137
.0210
.3223
.0214
.3309
17
.0218
.3394
.0223
.3478
.0227
.3561
.0231
.3644
.0236
.3726
.0240
.3806
18
.0245
.3887
.0249
.3966
.0254
.4045
.0258
.4123
.0263
.4200
.0268
.4276
19
.0272
.4352
.0277
.4427
.0282
.4502
.0287
.4576
.0292
.4649
.0297
.4721
20
.0302
.4793
.0307
.4865
.0312
.4936
.0317
.5006
.0322
.5075
.0327
.5144
21
.0332
.5213
.0337
.5281
.0343
.5348
.0348
.5415
.0353
.5481
.0359
.5547
22
.0364
.5612
.0370
.5677
.0375
.5741
.0381
.5805
.0386
.5868
.0392
.5931
23
.0397
.5993
.0403
.6055
.0409
.6116
.0415
.6177
.0421
.6238
.0426
.6298
24
.0432
.6357
.0438
.6417
.0444
.6476
.0450
.6534
.0456
.6592
.0462
.6650
25
.0468
.6707
.0475
.6764
.0481
.6820
.0487
.6876
.0493
.6932
.0500
.6987
26
.0506
.7042
.0512
.7096
.0519
.7151
.0525
.7204
.0532
.7258
.0538
.7311
27
.0545
.7364
.0552
.7416
.0558
.7468
.0565
.7520
.0572
.7572
.0578
.7623
28
.0585
.7673
.0592
.7724
.0599
.7774
.0606
.7824
.0613
.7874
.0620
.7923
29
.0627
.7972
.0634
.8020
.0641
.8069
.0648
.8117
.0655
.8165
.0663
.8213
30
.0670
.8260
.0677
.8307
.0684
.8354
.0692
.8400
.0699
.8446
.0707
.8492
31
.0714
.8538
.0722
.8583
.0729
.8629
.0737
.8673
.0744
.8718
.0752
.8763
32
.0760
.8807
.0767
.8851
.0775
.8894
.0783
.8938
.0791
.8981
.0799
.9024
33
.0807
.9067
.0815
.9109
.0823
.9152
.0831
.9194
.0839
.9236
.0847
.9277
34
.0855
.9319
.0863
.9360
.0871
.9401
.0879
.9442
.0888
.9482
.0896
.9523
35
.0904
.9563
.0913
.9603
.0921
.9643
.0929
.9682
.0938
.9722
.0946
.9761
36
.0955
.9800
.0963
.9838
.0972
.9877
.0981
.9915
.0989
.9954
.0998
.9992
37
.1007
.0030
.1016
.0067
.1024
.0105
.1033
.0142
.1042
.0179
.1051
.0216
38
.1060
.0253
.1069
.0289
.1078
.0326
.1087
.0362
.1096
.0398
.1105
.0434
39
.1114
.0470
.1123
.0505
.1133
.0541
.1142
.0576
.1151
.0611
.1160
.0646
40
.1170
.0681
.1179
.0716
.1189
.0750
.1198
.0784
.1207
.0817
.1217
.0853
41
.1226
.0887
.1236
.0920
.1246
.0954
.1255
.0987
.1265
.1021
.1275
.1054
42
.1284
.1087
.1294
.1119
.1304
.1152
.1314
.1185
.1323 w
.1217
.1333
.1249
43
.1343
.1282
.1353
.1314
.1363
.1345
.1373
.1377
.1383'. 1409
.1393
.1440
44
.1403
.1472
.1413
.1503
.1424
.1534
.1434
.1565
.1444
.1596
.1454
.1626
45
.1464
.1657
.1475
.1687
.1485
.1718
.1495
.1748
.1506
.1778
.1516
.1808
46
.1527
.1838
.1538
.1867
.1548
.1897
.1558
.1926
.1569
.1956
.1579
.1985
47
.1590
.2014
.1600
.2043
.1611
.2072
.1622
.2101
.1633
.2129
.1644
.2158
48
.1654
.2186
.1665
.2215
.1676
.2243
.1687
.2271
.1698
.2299
.1709
.2327
49
.1720
.2355
1731
.2382
.1742
.2410
.1753
.2437
.1764
.2465
.1775
.2492
50
.1786
.2519
.1797
.2546
.1808
.2573
.1820
.2600
.1831
.2627
.1842
.2653
51
.1853
.2680
.1865
.2700
.1876
.2732
.1887
.2759
.1899
.2785
.1910
.2811
52
.1922
.2837
.1933
.2863
.1945
.2888
.1956
.2914
.1968
.2940
.1979
.2965
53
.1991
.2991
.2003
.3016
.2014
.3041
.2026
.3066
.2038
:3091
.2049
.3116
54
.2061
.3141
.2073
.3166
.2085
.3190
.2096
.3215
.2108
.3239
.2120
.3264
55
.2132
.3288
.2144
.3312
.2156
.3336
.2168
.3361
.2180
.3384
.2192
.3408
56
.2204
.3432
.2216
.3456
.2228
.3480
.2240
.3503
.2252
.3527
.2265
.3550
57
.2277
.3573
.2289
.3596
.2301
.3620
.2314
.3643
.2326
.3666
.2338
.3689
58
.2350
.3711
.2363
.3734
.2375
.3757
.2388
.3779
.2400
.3802
.2412
.3824
59
.2425
.3847  .2437
.3869
.2450
.3891
.2462
.3913
.2475
.3935
.2487
.3957
118 Values and Logarithms of Haversines
[Characteristics of Logarithms omitted — determine by rule from the value]
60
61
62
63
64
65
66
67
68
69
70
71
72
78
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
0'
Value Log 10
.2500
.2576
.2653
.2730
.2808
.2887
.2966
.3046
.3127
.3208
.3290
.3372
.3455
.3538
.3622
.3706
.3790
.3875
.3960
.4046
.4132
.4218
.4304
.4391
.4477
.4564
.4651
.4738
.4826
.4913
.5000
.5087
.5174
.5262
.5349
.5436
.5523
.5609
.5696
.5782
.5868
.5954
.6040
.6125
.6210
.6294
.6378
.6462
.6545
.6628
.6710
.6792
.6873
.6954
.7034
.7113
.7192
.7270
.7347
.7424
.3979
.4109
.4237
.4362
.4484
.4604
.4722
.4838
.4951
.5063
.5172
.5279
.5384
.5488
.5589
.5689
.5787
.5883
.5977
.6070
.6161
.6251
.6339
.6425
.6510
.6594
.6676
.6756
.6835
.6913
.6990
.7065
.7139
.7211
.7283
.7353
.7421
.7489
.7556
.7621
.7685
.7748
.7810
.7871
.7931
.7989
.8047
.8104
.8159
.8214
.8267
.8320
.8371
.8422
.8472
.8521
.8568
.8615
.8661
.8706
10'
Value Log 10
.2513
.2589
.2665
.2743
.2821
.2900
.2980
.3060
.3140
.3222
.3304
.3386
.3469
.3552
.3636
.3720
.3805
.3889
.3975
.4060
.4146
.4232
.4319
.4405
.4492
.4579
.4753
.4840
.4937
.5015
.5102
.5189
.5276
.5363
.5450
.5537
.5624
.5710
.5797
.5883
.5968
.6054
.6139
.6224
.6308
.6392
.6476
.6559
.6642
.6724
.6805
.6887
.6967
.7047
.7126
.7205
.7283
.7360
.7437
.4001
.4131
.4258
.4382
.4504
.4624
.4742
.4857
.4970
.5081
.5190
.5297
.5402
.5505
.5606
.5705
.5803
.5899
.5993
.6085
.6176
.6266
.6353
.6440
.6524
.6607
.6689
.6770
.6848
.6926
.7002
.7077
.7151
.7223
.7294
.7364
.7433
.7500
.7567
.7632
.7696
.7759
.7820
.7881
.7940
.7999
.8056
.8113
.8168
.8223
.8276
.8329
.8380
.8430
.8480
.8529
.8576
.8623
.8669
.8714
20'
Value Log t
30'
Value Log 10
.2525 .4023
.2601 .4152
.2678 .4279
.2756 .4403
.2834 .4524
.2913 .4644
.2993 .4761
.3073 .4876
.3154 .4989
.3235 .5099
.3317 .5208
.3400 .5314
.3483 .5419
.3566 .5522
.3650 .5623
.3734 .5722
.3819 .5819
.3904 .5915
.3989 .6009
.4075 .6101
.4160 .6191
.4247 .6280
.4333 .6368
.4420 .6454
.4506 .6538
.4593 .6621
.4680 .6703
.4767 .6783
.4855 .6862
.4942 .6939
.5029 .7015
.5116 .7090
.5204 .7163
.5291 .7235
.5378 .7306
.5465
.5552
.5638
.5725
.5811
.5897
.5983
.6068
.6153
.6238
.6322
.6406
.6490
.6573
.6655
.6737
.6819
.6900
,6980
.7060
.7139
.7218
.7296
.7373
.7449
.7376
.7444
.7511
.7577
.7642
.7706
.7769
.7830
.7891
.7950
.8009
.8122
.8177
.8232
.8285
.8337
.8388
.8439
.8488
.8537
.8584
.8631
.8676
.8721
.2538
.2614
.2691
.2769
.2847
.2927
.3006
.3087
.3167
.3249
.3331
.3413
.3496
.3580
.3664
.3748
.3833
.3918
.4003
.4089
.4175
.4261
.4347
.4434
.4521
.4608
.4695
.4782
.4869
.4956
.5044
.5131
.5218
.5305
.5392
.5479
.5566
.5653
.5739
.5825
.5911
.5997
.6082
.6167
.6252
.6336
.6420
.6504
.6587
.6669
.6751
.6833
.6913
.6994
.7073
.7153
.7231
.7309
.7386
.7462
.4045
.4173
.4300
.4423
.4545
.4664
.4780
.4895
.5007
.5117
.5226
.5332
.5436
.5539
.5639
.5738
.5835
.5930
.6024
.6116
.6206
.6295
.6382
.6468
.6552
.6635
.6716
.6796
.6875
.6952
.7027
.7102
.7175
.7247
.7318
.7387
.7455
.7523
.7588
.7653
.7717
.7779
.7841
.7901
.7960
.8018
.8075
.8131
.8187
.8241
.8294
.8346
.8397
.8447
.8496
.8545
.8592
.8638
.8684
.8729
40'
Value Log 10
.2551
.2627
.2704
.2782
.2861
.2940
.3020
.3100
.3181
.3263
.3345
.3427
.3510
.3594
.3678
.3762
.3847
.3932
.4017
.4103
.4189
.4275
.4362
.4448
.4535
.4622
.4709
.4796
.4884
.4971
.5058
.5145
.5233
.5320
.5407
.5494
.5580
.5667
.5753
.5840
.5925
.6011
.6096
.6181
.6266
.6350
.6434
.6517
.6600
.6683
.6765
.6846
.6927
.7007
.7087
.7166
.7244
.7322
.7399
.7475
.4066
.4195
.4320
.4444
.4565
.4683
.4799
.4914
.5026
.5136
.524*4
.5349
.5454
.5556
.5656
.5754
.5851
.5946
.6039
.6131
.6221
.6310
.6397
.6482
.6566
.6649
.6730
.6809
.6887
.6964
.7040
.7114
.7187
.7259
.7329
.7399
.7467
.7534
.7599
.7664
.7727
.7790
.7851
.7911
.7970
.8028
.8085
.8141
.8196
.8250
.8302
.8354
.8405
.8455
.8504
.8553
.8600
.8646
.8691
.8736
50'
Value Log 10
.2563
.2640
.2717
.2795
.2874
.2953
.3033
.3113
.3195
.3276
.3358
.3441
.3524
.3608
.3692
.3776
.3861
.3946
.4032
.4117
.4203
.4290
.4376
.4463
.4550
.4637
.4724
.4811
.4898
.4985
.5073
.5160
.5247
.5334
.5421
.5508
.5595
.5682
.5768
.5854
.5940
.6025
.6111
.6195
.6280
.6364
.6448
.6531
.6614
.6696
.6778
.6860
.6940
.7020
.7100
.7179
.7257
.7335
.7411
.7487
.4088
.4216
.4341
.4464
.4584
.4703
.4819
.4932
.5044
.5154
.5261
.5367
.5471
.5572
.5672
.5771
.5867
.5962
.6055
.6146
.6236
.6324
.6411
.6496
.6580
.6662
.6743
.6822
.6900
.6977
.7052
.7126
.7199
.7271
.7341
.7410
.7478
.7545
.7610
.7674
.7738
.7800
.7861
.7921
.7980
.8037
.8094
.8150
.8205
.8258
.8311
.8363
.8414
.8464
.8513
.8561
.8608
.8654
.8699
.8743
Values and Logarithms of Haversines
[Characteristics of Logarithms omitted — determine by rule from the value]
119
.
C
10'
20'
30'
40'
5)'
Value
Log w
Value
Logi
Value
Log 10
Value
Log 10
Value
Log 10
Value
Log 10
120
.7500
.8751
.7513
.8758
.7525
.8765
.7538
.8772
.7550
.8780
.7563
.8787
121
.7575
.8794
.7588
.8801
.7600
.8808
.7612
.8815
.7625
.8822
.7637
.8829
122
.7650
.8836
.7662
.8843
.7674
.8850
.7686
.8857
.7699
.8864
.7711
.8871
123
.7723
.8878
.7735
.8885
.7748
.8892
.7760
.8898
.7772
.8905
.7784
.8912
124
.7796
.8919
.7808
.8925
.7820
.8932
.7832
.8939
.7844
.8945
.7856
.8952
125
.7868
.8959
.7880
.8965
.7892
.8972
.7904
.8978
.7915
.8985
.7927
.8991
126
.7939
.8998
.7951
.9004
.7962
.9010
.7974
.9017
.7986
.9023
.7997
.9030
127
.8009
.9036
.8021
.9042
.8032
.9048
.8044
.9055
.8055
.9061
.8067
.9067
128
.8078
.9073
.8090
.9079
.8101
.9085
.8113
.9092
.8124
.9098
.8135
.9104
129
.8147
.9110
.8158
.9116
.8169
.9122
.8180
.9128
.8192
.9134
.8203
.9140
130
.8214
.9146
.8225
.9151
.8236
.9157
.8247
.9163
.8258
.9169
.8269
.9175
131
.8280
.9180
.8291
.9186
.8302
.9192
.8313
.9198
.8324
.9203
.8335
.9209
132
.8346
.9215
.8356
.9220
.8367
.9226
.8378
.9231
.8389
.9237
.8399
.9242
133
.8410
.9248
.8421
.9253
.8431
.9259
.8442
.9264
.8452
.9270
.8463
.9275
134
.8473
.9281
.8484
.9286
.8494
.9291
.8501
.9297
.8515
.9302
.8525
.93p7
135
.8536
.9312
.8546
.9318
.8556
.9323
.8566
.9328
.8576
.9333
.8587
.9338
136
.8597
.9343
.8607
.9348
.8617
.9353
.8627
.9359
.8637
.9364
.8647
.9369
137
.8657
.9374
.8667
.9379
.8677
.9383
.8686
.9388
.8696
.9393
.8706
.9398
138
.8716
.9403
.8725
.9408
.8735
.9413
.8745
.9417
.8754
.9422
.8764
.9427
139
.8774
.9432
.8783
.9436
.8793
.9441
.8802
.9446
.8811
.9450
.8821
.9455
140
.8830
.9460
.8840
.9464
.8849
.9469
.8858
.9473
.8867
.9478
.8877
.9482
141
.8886
.9487
.8895
.9491
.8904
.9496
.8913
.9500
.8922
.9505
.8931
.9509
142
.8940
.9513
.8949
.9518
.8958
.9522
.8967
.9526
.8976
.9531
.8984
.9535
143
.8993
.9539
.9002
.9543
.9011
.9548
.9019
.9552
.9028
.9556
.9037
.9560
144
.9045
.9564
.9054
.9568
.9062
.9572
.9071
.9576
.9079
.9580
.9087
.9584
145
.9096
.9588
.9104
.9592
.9112
.9596
.9121
.9600
.9129
.9604
.9137
.9608
146
.9145
.9612
.9153
.9616
.9161
.9620
.9169
.9623
.9177
.9627
.9185
.9631
147
.9193
.9635
.9201
.9638
.9209
.9642
.9217
.9646
.9225
.9650
.9233
.9653
148
.9240
.9657
.9248
.9660
.9256
.9664
.9263
.9668
.9271
.9671
.9278
.9675
149
.9286
.9678
.9293
.9682
.9301
.9685
.9308
.9689
.9316
.9692
.9323
.9695
150
.9330
.9699
.9337
.9702
.9345
.9706
.9352
.9709
.9359
.9712
.9366
.9716
151
.9373
.9719
.9380
.9722
.9387
.9725
.9394
.9729
.9401
.9732
.9408
.9735
152
.9415
.9738
.9422
.9741
.9428
.9744
.9435
.9747
.9442
.9751
.9448
.9754
153
.9455
.9757
.9462
.9760
.9468
.9763
.9475
.9766
.9481
.9769
.9488
.9772
154
.9494
.9774
.9500
.9777
.9507
.9780
.9513
.9783
.9519
.9786
.9525
.9789
155
.9532
.9792
.9538
.9794
.9544
.9797
.9550
.9800
.9556
.9803
.9562
.9805
156
.9568
.9808
.9574
.9811
.9579
.9813
.9585
.9816
.9591
.9819
.9597
.9821
157
.9603
.9824
.9608
.9826
.9614
.9829
.9619
.9831
.9625
.9834
.9630
.9836
158
.9636
.9839
.9641
.9841
.9647
.9844
.9652
.9846
.9657
.9849
.9663
.9851
159
.9668
.9853
.9673
.9856
.9678
.9858
.9683
.9860
.9688
.9863
.9693
.9865
160
.9698
.9867
.9703
.9869
.9708
.9871
.9713
.9874
.9718
.9876
.9723
.9878
161
.9728
.9880
.9732
.9882
.9737
.9884
.9742
.9886
.9746
.9888
.9751
.9890
162
.9755
.9892
.9760
.9894
.9764
.9896
.9769
.9898
.9773
.9900
.9777
.9902
163
.9782
.9904
.9786
.9906
.9790
.9908
.9794
.9910
.9798
.9911
.9802
.9913
164
.9806
.9915
.9810
.9917
.9814
.9919
.9818
.9920
.9822
.9922
.9826
.9923
165
.9830
.9925
.9833
.9927
.9837
.9929
.9841
.9930
.9844
.9932
.9848
.9933
166
.9851
.9935
.9855
.9937
.9858
.9938
.9862
.9940
.9865
.9941
.9869
.9943
167
.9872
.9944
.9875
.9945
.9878
.9947
.9881
.9948
.9885
.9950
.9888
.9951
168
.9891
.9952
.9894
.9954
.9897
.9955
.9900
.9956
.9903
.9957
.9905
.9959
169
.9908
.9960
.9911
.9961
.9914
.9962
.9916
.9963
.9919
.9965
.9921
.9966
170
.9924
.9967
.9927
.9968
.9929
.9969
.9931
.9970
.9934
.9971
.9936
.9972
171
.9938
.9973
.9941
.9974
.9943
.9975
.9945
.9976
.9947
.9977
.9949
9978
172
.9951
.9979
.9953
.9980
.9955
.9981
.9957
.9981
.9959
.9982
.9961
.9983
173
.9963
.9984
.9964
.9984
.9966
.9985
.9968
.9986
.9969
.9987
.9971
.9987
174
.9973
.9988
.9974
.9988
.9976
.9989
.9977
.9990
.9978
.9991
.9980
.9991
175
.9981
.9992
.9982
.9992
.9983
.9993
.9985
.9993
.9986
.9994
.9987
.9994
176
.9988
.9995
.9989
.9995
.9990
.9996
.9991
.9996
.9992
.9996
.9992
.9997
177
.9993
.9997
.9994
.9997
.9995
.9998
.9995
.9998
.9996
.9998
.9996
.9998
178
.9997
.9999
.9997
.9999
.9998
.9999
.9998
.9999
.9999
.9999
.9999
.9999
179
.9999
.9999
.9999
.9999
.9999
.9999
.9999
.9999
.9999 0.0000
1.0000
.0000
INDEX
Abscissa, 6.
Absolute value, of a directed quan
tity, 7.
Addition, of angles, 9; formulas in
trigonometry, 95.
Angle, definition of, 7; directed, 7;
measurement of, 8 ; addition and
subtraction of, 9 ; functions of, 2 ;
of elevation and depression, 16;
of triangle, 48 ; in artillery service,
76.
Annuities, 70.
Arc of a circle, 76.
Artillery service, use of angles in, 76.
Axes, of coordinates, 5.
Briggian logarithms, 54.
Characteristic of a logarithm, 54.
Cologarithms, 59.
Common logarithms, 54.
Compass, Mariner's, 29.
Computation, numerical,
logarithmic, 61 ff.
Coordinates in a plane, 5.
Cosecant, 32.
Cosine, definition of, 12 :
of, 81 ; graph of, 82 ;
40.
Cotangent, definition of, 32
Course, 29.
Coversed sine, 32.
18, 24
; variation
law of — s,
Dead reckoning, 30.
Departure, 29.
Difference in latitude, 29 ; in longi
tude, 30.
Directed, angles, 7 ; quantities, 6
segments, 7.
Distance, 29.
Elements of a triangle, 1.
Function, definition of, 3 ; representa
tion of, 32 ; trigonometric, 12 ff .,
58.
Graph of trigonometric functions,
80, 82, 83.
Haversine, definition of, 32; solu
tion of triangles by, 48 ; tables of,
1179.
Identities, trigonometric, 35.
Initial position, 7.
Interest, 70.
Interpolation, 22.
Knot, 29.
Latitude, difference in, 29 ; middle,
30.
Law, of sines, 40 ; cosines, 40 ; of
tangents, 47.
Logarithm, definition of, 52 ; inven
tion of, 50 ; laws of, 53 ; systems
of, 54 ; characteristic and man
tissa of, 54 ; use of tables of, 56 ;
tables of, 11016.
Logarithmic scale, 73.
Magnitude, 6.
Mantissa, 54.
Mariner's compass, 29.
Middle latitude, 30.
Mil, 76.
Napier, J., 50.
Nautical mile, 29.
Navigation, 28 ff .
Negative angle, definition of, 7;
functions of, 85.
Ordinate, 6.
121
122
INDEX
Parts of a triangle, 1.
Period of trigonometric functions,
80, 82, 84.
Plane sailing, 28.
Plane trigonometry, 1.
Product formulas, 101.
Projectile, 72.
Projection, 92.
Quadrant, 6.
Radian, 75.
Radius of inscribed circle, 46.
Rotation, angles of, 8.
Rounded numbers, 25.
Scale, logarithmic, 73.
Secant, definition of, 32.
Significant figures, 25.
Sine, definition of, 12 ; variation of,
79 ; graph of, 80 ; law of s, 40.
Slide rule, 74.
Solution of triangles, 1, 16 ff., 41 ft*.,
48, 62 ff.
Spherical trigonometry, 1.
Tables, of squares, 27, 1067; of
haversines, 1179; of logarithms,
11011 ; of trigonometric func
tions, 11219.
Tangent, definition of, 3, 12 ; variation
of, 82 ; graph of, 83 ; line repre
sentation of, 83 ; law of s, 47.
Triangle, area of, 45 ; angles of, 48 ;
solution of, 1, 16 ff., 41 ff., 48, 62.
Trigonometric equations, 88.
Trigonometric functions, definitions
of, 3, 12, 15, 32 ; graphs of, 80, 82,
83 ; computation of, 18 ff . ; periods
of, 80, 82, 84; inverse, 87; formulas,
15, 32, 34, 96 ff. ; logarithms of,
61 ; tables of, 21, 11219.
Versed sine, defined, 32.
Printed in the United States of America.
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ELEMENTARY MATHEMATICAL
ANALYSIS
BY
JOHN WESLEY YOUNG
Professor of Mathematics in Dartmouth College
And FRANK MILLET MORGAN
Assistant Professor of Mathematics in Dartmouth College
Edited by Earle Raymond Hedrick, Professor of Mathematics
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Cloth, 8vo, $1.25
The chief aims of this text are brevity, clarity, and simplicity. The author presents the
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CONTENTS
PLANE TRIGONOMETRY chapter
6. The Solution of General Triangles . .
The Solution of Trigonometric Equa
CHAPTER
1. The Trigonometric Functions of Any tions
Angle and Identical Relations among
Them
2. Identical Relations Among the Func SPHERICAL TRIGONOMETRY
tions of Related Angles: The Values
of the Functions of Certain Angles 8. Fundamental Relations
3. The Solution of Right Triangles. 9. The Solution of Right Spherical Tri
Logarithms and Computation by angles
Means of Logarithms 10. The Solution of Oblique Spherical
4. Fundamental Identities Triangles
5. The Circular or Radian Measure of an n. The Earth as a Sphere ...
Angle. Inverse Trigonometric Func Answers
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Differential and Integral Calculus
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Crown 8vo, $2.10
Presents a first course in the calculus — substantially as the author has
taught it at the University of Michigan for a number of years. The follow
ing points may be mentioned as more or less prominent features of the book :
In the treatment of each topic the author has presented his material in
such a way that he focuses the student's attention upon the fundamental
principle involved, insuring his clear understanding of that, and preventing
him from being confused by the discussion of a multitude of details. His
constant aim has been to prevent the work from degenerating into mere
mechanical routine; thus, wherever possible, except in the purely formal
parts of the course, he has avoided the summarizing of the theory into
rules or formulas which can be applied blindly.
The Calculus
By ELLERY WILLIAMS DAVIS
Professor of Mathematics, the University of Nebraska
Assisted by William Charles Brenke, Associate Professor of Mathe
matics, the University of Nebraska
Edited by Earle Raymond Hedrick
Cloth, semi flexible, with Tables, i2tno, $2.10
Edition De Luxe, flexible leather binding, $2.50
This book presents as many and as varied applications of the Calculus
as it is possible to do without venturing into technical fields whose subject
matter is itself unknown and incomprehensible to the student, and without
abandoning an orderly presentation of fundamental principles.
The same general tendency has led to the treatment of topics with a view
toward bringing out their essential usefulness. Rigorous forms of demon
stration are not insisted upon, especially where the precisely rigorous proofs
would be beyond the present grasp of the student. Rather the stress is laid
upon the student's certain comprehension of that which is done, and his con
viction that the results obtained are both reasonable and useful. At the
same time, an effort has been made to avoid those grosser errors and actual
misstatements of fact which have often offended the teacher in texts other
wise attractive and teachable.
THE MACMILLAN COMPANY
Publishers 6466 Fifth Avenue New Tork
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THE UNIVERSITY OF CALIFORNIA LIBRARY
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